Introduction to Statistical Analysis

The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.

In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct.

Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life.

Probability

Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is \(\frac{1}{2}\) or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction \(\frac{996}{2000}\) is equal to 0.498 which is very close to 0.5, the expected probability.

The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.

Key Terms

In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.

Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000–2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of carbonated drink.

From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter, in this case the mean. A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.

One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.

A variable, or random variable, usually notated by capital letters such as \(x\) and \(y\), is a characteristic or measurement that can be determined for each member of a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let \(x\) equal the number of points earned by one math student at the end of a term, then \(x\) is a numerical variable. If we let \(y\) be a person's party affiliation, then some examples of \(y\) include Republican, Democratic, and Independent. \(y\) is a categorical variable. We could do some math with values of  \(x\) (calculate the average number of points earned, for example), but it makes no sense to do math with values of \(y\) (calculating an average party affiliation makes no sense).

Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.

Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is \(\frac{22}{40}\) and the proportion of women students is \(\frac{18}{40}\). Mean and proportion are discussed in more detail in later chapters.

EXAMPLE 1.1

Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively.

Solution 1.1

The population is all first year students attending ABC College this term.

The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population).

The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term: the population mean.

The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample.

The variable could be the amount of money spent (excluding books) by one first year student. Let \(x=\) the amount of money spent (excluding books) by one first year student attending ABC College.

The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225.

TRY IT 1.1

Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey 100 families with children in the school. Three of the families spent $65, $75, and $95, respectively.

EXAMPLE 1.2

Determine what the key terms refer to in the following study.
A study was conducted at a local college to analyze the average cumulative GPA's of students who graduated last year. Fill in the letter of the phrase that best describes each of the items below.

1. Population ____ 2. Statistic ____ 3. Parameter ____ 4. Sample ____ 5. Variable ____ 6. Data ____

1. all students who attended the college last year
2. the cumulative GPA of one student who graduated from the college last year
3. 3.65, 2.80, 1.50, 3.90
4. a group of students who graduated from the college last year, randomly selected
5. the average cumulative GPA of students who graduated from the college last year
6. all students who graduated from the college last year
7. the average cumulative GPA of students in the study who graduated from the college last year

Solution 1.2
1. f; 2. g; 3. e; 4. d; 5. b; 6. c

EXAMPLE 1.3

Determine what the key terms refer to in the following study. As part of a study designed to test the safety of automobiles, the National Transportation Safety Board collected and reviewed data about the effects of an automobile crash on test dummies. Here is the criterion they used:

Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver's seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars.

Solution 1.3

The population is all cars containing dummies in the front seat.

The sample is the 75 cars, selected by a simple random sample.

The parameter is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population.

The statistic is proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample.

The variable \(x =\) the number of driver dummies (if they had been real people) who would have suffered head injuries.

The data are either: yes, had head injury, or no, did not.

EXAMPLE 1.4

Determine what the key terms refer to in the following study.

An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.

Solution 1.4

The population is all medical doctors listed in the professional directory.

The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population.

The sample is the 500 doctors selected at random from the professional directory.

The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample.

The variable \(x=\) the number of medical doctors who have been involved in one or more malpractice suits.

The data are either: yes, was involved in one or more malpractice lawsuits, or no, was not.

Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem.

Figure 2.2

Solution 2.5

Solution 2.6

Figure 2.4

NOTE
We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.

The boundaries are:

• 59.95
• 59.95 + 2 = 61.95
• 61.95 + 2 = 63.95
• 63.95 + 2 = 65.95
• 65.95 + 2 = 67.95
• 67.95 + 2 = 69.95
• 69.95 + 2 = 71.95
• 71.95 + 2 = 73.95
• 73.95 + 2 = 75.95

The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.

The following histogram displays the heights on the x-axis and relative frequency on the y-axis.

Figure 2.5

\(\frac{6.5−0.5}{ <code>\textrm </code>number of bars}=1\)

where 1 is the width of a bar. Therefore, bars = 6.

The following histogram displays the number of books on the x-axis and the frequency on the y-axis.

Figure 2.6

Solution 2.10

Figure 2.7

Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.

Frequency distribution for calculus final test scores
Lower bound	Upper bound	Frequency	Cumulative frequency
49.5	59.5	5	5
59.5	69.5	10	15
69.5	79.5	30	45
79.5	89.5	40	85
89.5	99.5	15	100

Table 2.14

Figure 2.8

The first label on the x-axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x-axis. The point labeled 54.5 represents the next interval, or the first "real" interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Frequency distribution for calculus final grades
Lower bound	Upper bound	Frequency	Cumulative frequency
49.5	59.5	10	10
59.5	69.5	10	20
69.5	79.5	30	50
79.5	89.5	45	95
89.5	99.5	5	100

Table 2.17

Figure 2.9

Constructing a Time Series Graph

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

Solution 2.13

Figure 2.10

The common measures of location are quartiles and percentiles

Quartiles are special percentiles. The first quartile, Q₁, is the same as the 25^th percentile, and the third quartile, Q₃, is the same as the 75^th percentile. The median, M, is called both the second quartile and the 50^th percentile.

To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the 90^th percentile of an exam does not mean, necessarily, that you received 90% on a test. It means that 90% of test scores are the same or less than your score and 10% of the test scores are the same or greater than your test score.

Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75^th percentile. That translates into a score of at least 1220.

Percentiles are mostly used with very large populations. Therefore, if you were to say that 90% of the test scores are less (and not the same or less) than your score, it would be acceptable because removing one particular data value is not significant.

The median is a number that measures the "center" of the data. You can think of the median as the "middle value," but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data.
1; 11.5; 6; 7.2; 4; 8; 9; 10; 6.8; 8.3; 2; 2; 10; 1
Ordered from smallest to largest:
1; 1; 2; 2; 4; 6; 6.8; 7.2; 8; 8.3; 9; 10; 10; 11.5

Since there are 14 observations, the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two.

\(\frac{6.8+7.2}{2} =7\)

The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.

Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median or second quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q₃, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:
1; 1; 2; 2; 4; 6; 6.8; 7.2; 8; 8.3; 9; 10; 10; 11.5

The median or second quartile is seven. The lower half of the data are 1, 1, 2, 2, 4, 6, 6.8. The middle value of the lower half is two.
1; 1; 2; 2; 4; 6; 6.8

The number two, which is part of the data, is the first quartile. One-fourth of the entire sets of values are the same as or less than two and three-fourths of the values are more than two.

The upper half of the data is 7.2, 8, 8.3, 9, 10, 10, 11.5. The middle value of the upper half is nine.

The third quartile, Q₃, is nine. Three-fourths (75%) of the ordered data set are less than nine. One-fourth (25%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example.

The interquartile range is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile (Q₃) and the first quartile (Q₁).

IQR = Q₃ – Q₁

The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile. Potential outliers always require further investigation.

NOTE

Example 2.14

For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. Prices are in dollars.
389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000

Solution 2.14

Example 2.15

For the two data sets in the test scores example, find the following:

    a. The interquartile range. Compare the two interquartile ranges.
    b. Any outliers in either set.

The five number summary for the day and night classes is

Table 2.21

Example 2.16

Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were:

Find the 28th percentile. Notice the 0.28 in the "cumulative relative frequency" column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28^th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28^th percentile is between the last six and the first seven. The 28th percentile is 6.5.

Find the median. Look again at the "cumulative relative frequency" column and find 0.52. The median is the 50^th percentile or the second quartile. 50% of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and eleven of the 7s. The median or 50^th percentile is between the 25^th, or seven, and 26^th, or seven, values. The median is seven.

Find the third quartile. The third quartile is the same as the 75^th percentile. You can "eyeball" this answer. If you look at the "cumulative relative frequency" column, you find 0.52 and 0.80. When you have all the fours, fives, sixes and sevens, you have 52% of the data. When you include all the 8s, you have 80% of the data. The 75^th percentile, then, must be an eight. Another way to look at the problem is to find 75% of 50, which is 37.5, and round up to 38. The third quartile, Q₃, is the 38^th value, which is an eight. You can check this answer by counting the values. (There are 37 values below the third quartile and 12 values above).

Try It 2.16

Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65^th percentile.

Example 2.17

Using Table 2.22:

   a. Find the 80th percentile.
   b. Find the 90th percentile.
   c. Find the first quartile. What is another name for the first quartile?

b. The 90^th percentile will be the 45^th data value (location is 0.90(50) = 45) and the 45^th data value is nine.

c. Q1 is also the 25^th percentile. The 25^th percentile location calculation: P25 = 0.25(50) = 12.5 ≈ 13 the 13^th data value. Thus, the 25^th percentile is six.

A Formula for Finding the k^th Percentile

If you were to do a little research, you would find several formulas for calculating the k^th percentile. Here is one of them.

\(k\) = the k^th percentile. It may or may not be part of the data.

\(i\) = the index (ranking or position of a data value)

\(n\) = the total number of data points, or observations

• Order the data from smallest to largest.
• Calculate \(i=\frac{k}{100} (n+1)\)
• If \(i\) is an integer, then the k^th percentile is the data value in the i^th position in the ordered set of data.
• If \(i\) is not an integer, then round \(i\) up and round \(i\) down to the nearest integers. Average the two data values in these two positions in the ordered data set. This is easier to understand in an example.
  

Example 2.18

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.
18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77

  a. Find the 70^th percentile.
  b. Find the 83^rd percentile.

Try It 2.18

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.

18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77
Calculate the 20^th percentile and the 55^th percentile.

A Formula for Finding the Percentile of a Value in a Data Set

• Order the data from smallest to largest.
• \(x =\) the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile.
• \(y =\) the number of data values equal to the data value for which you want to find the percentile.
• \(n =\) the total number of data.
• Calculate \(\frac{x+0.5y}{n}(100)\). Then round to the nearest integer.
  

Example 2.19

Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.
18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77

Interpreting Percentiles, Quartiles, and Median

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the p^th percentile. For example, 15% of data values are less than or equal to the 15^th percentile.

• Low percentiles always correspond to lower data values.
• High percentiles always correspond to higher data values.

A percentile may or may not correspond to a value judgment about whether it is "good" or "bad." The interpretation of whether a certain percentile is "good" or "bad" depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered "good;" in other contexts a high percentile might be considered "good". In many situations, there is no value judgment that applies.

Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.

NOTE

• information about the context of the situation being considered
• the data value (value of the variable) that represents the percentile
• the percent of individuals or items with data values below the percentile
• the percent of individuals or items with data values above the percentile.

Example 2.20

On a timed math test, the first quartile for time it took to finish the exam was 35 minutes. Interpret the first quartile in the context of this situation.

Solution 2.20

• Twenty-five percent of students finished the exam in 35 minutes or less.
• Seventy-five percent of students finished the exam in 35 minutes or more.
• A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. (If you take too long, you might not be able to finish).

Example 2.21

On a 20 question math test, the 70^th percentile for number of correct answers was 16. Interpret the 70^th percentile in the context of this situation.

Solution 2.21

• Seventy percent of students answered 16 or fewer questions correctly.
• Thirty percent of students answered 16 or more questions correctly.
• A higher percentile could be considered good, as answering more questions correctly is desirable.

Try It 2.21

On a 60 point written assignment, the 80^th percentile for the number of points earned was 49. Interpret the 80^th percentile in the context of this situation.

Example 2.22

At a community college, it was found that the 30^th percentile of credit units that students are enrolled for is seven units. Interpret the 30^th percentile in the context of this situation.

Solution 2.22

• Thirty percent of students are enrolled in seven or fewer credit units.
• Seventy percent of students are enrolled in seven or more credit units.
• In this example, there is no "good" or "bad" value judgment associated with a higher or lower percentile. Students attend community college for varied reasons and needs, and their course load varies according to their needs.

Example 2.23

Sharpe Middle School is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed 15 anonymous students to determine how many minutes a day the students spend exercising. The results from the 15 anonymous students are shown.

0 minutes; 40 minutes; 60 minutes; 30 minutes; 60 minutes

10 minutes; 45 minutes; 30 minutes; 300 minutes; 90 minutes;

30 minutes; 120 minutes; 60 minutes; 0 minutes; 20 minutes

Determine the following five values.

• Min = 0
• Q₁ = 20
• Med = 40
• Q₃ = 60
• Max = 300
  

If you were the principal, would you be justified in purchasing new fitness equipment? Since 75% of the students exercise for 60 minutes or less daily, and since the IQR is 40 minutes (60 – 20 = 40), we know that half of the students surveyed exercise between 20 minutes and 60 minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment.

However, the principal needs to be careful. The value 300 appears to be a potential outlier.

Q₃ + 1.5(IQR) = 60 + (1.5)(40) = 120.

The value 300 is greater than 120 so it is a potential outlier. If we delete it and calculate the five values, we get the following values:

• Min = 0
• Q₁ = 20
• Q₃ = 60
• Max = 120
  

We still have 75% of the students exercising for 60 minutes or less daily and half of the students exercising between 20 and 60 minutes a day. However, 15 students is a small sample and the principal should survey more students to be sure of his survey results.

The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. Technically this is the arithmetic mean. We will discuss the geometric mean later. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts meaning an equal number of observations on each side. The weight of 25 people are below this weight and 25 people are heavier than this weight. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

NOTE

\(\overline x =\frac{1+1+1+2+2+3+4+4+4+4+4}{11}=2.7\)

\(\overline x =\frac{3(1)+2(2)+1(3)+5(4)}{11}=2.7\)

In the second calculation, the frequencies are 3, 2, 1, and 5.

You can quickly find the location of the median by using the expression \(\frac{n+1}{2}\).

The letter \(n\) is the total number of data values in the sample. If \(n\) is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If \(n\) is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is 97, then \(\frac{n+1}{2}= \frac{97+1}{2} = 49\). The median is the 49^th value in the ordered data. If the total number of data values is 100, then \(\frac{n+1}{2}= \frac{100+1}{2} = 50.5\). The median occurs midway between the 50^th and 51^st values. The location of the median and the value of the median are not the same. The upper case letter M is often used to represent the median. The next example illustrates the location of the median and the value of the median.

Example 2.24

---

AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):
3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47;
Calculate the mean and the median.

Solution 2.24

The calculation for the mean is:

\(\overline x=\frac{[3+4+(8)(2)+10+11+12+13+14+(15)(2)+(16)(2)+...+35+37+40+(44)(2)+47]}{40}=23.6\)

To find the median, M, first use the formula for the location. The location is:

\(\frac{n+1}{2}=\frac{40+1}{2}=20.5\)

Starting at the smallest value, the median is located between the 20th and 21st values (the two 24s):

3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47;

\(M=\frac{24+24}{2}=24\)

Example 2.25

---

Suppose that in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn $30,000. Which is the better measure of the "center": the mean or the median?

Solution 2.25

\(\overline x =\frac{5,000,000+49(30,000)}{50}=129,400\)

\(M = 30,000\)

(There are 49 people who earn $30,000 and one person who earns $5,000,000.)

The median is a better measure of the "center" than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data.

Another measure of the center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.

Example 2.26

---

Statistics exam scores for 20 students are as follows:

50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72; 76; 78; 81; 83; 84; 84; 84; 90; 93

Find the mode.

Solution 2.26

The most frequent score is 72, which occurs five times. Mode = 72.

Example 2.27

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Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice.

When is the mode the best measure of the "center"? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.

NOTE
The mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red.

Calculating the Arithmetic Mean of Grouped Frequency Tables

When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: \(\textrm
mean = \frac{\textrm data \: sum}{\textrm number \: of \: data \: values}\) We simply need to modify the definition to fit within the restrictions of a frequency table.

Since we do not know the individual data values we can instead find the midpoint of each interval. The midpoint is \(\frac{ \textrm lower \: boundary \: + \: upper \: boundary}{2}\) . We can now modify the mean definition to be \(\textrm Mean \: of \: Frequency \:  Table=\frac{∑fm}{∑f}\) where \(f =\) the frequency of the interval and \(m =\) the midpoint of the interval.

Example 2.28

---

A frequency table displaying professor Blount's last statistic test is shown. Find the best estimate of the class mean.

Solution 2.28

• Find the midpoints for all intervals
  
  

  Grade interval	Midpoint
  50–56.5	53.25
  56.5–62.5	59.5
  62.5–68.5	65.5
  68.5–74.5	71.5
  74.5–80.5	77.5
  80.5–86.5	83.5
  86.5–92.5	89.5
  92.5–98.5	95.5

  
  Table 2.25
  

• Calculate the sum of the product of each interval frequency and midpoint. \(∑​f 53.25(1)+59.5(0)+65.5(4)+71.5(4)+77.5(2)+83.5(3)+89.5(4)+95.5(1)=1460.25\)
  
  
• \(μ=\frac{∑fm}{∑f}=\frac{1460.25}{19}=76.86\)

Try It 2.28

---

Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:

What is the best estimate for the mean number of hours spent playing video games?

Formula for Population Mean

\(\mu=\frac{1}{N} \sum_{i=1}^{N} x_{i}\)

Formula for Sample Mean

\(\overline x =\frac{1}{n} ∑ _{i=1} ^{n} x_i\)

This unit is here to remind you of material that you once studied and said at the time "I am sure that I will never need this!"

Here are the formulas for a population mean and the sample mean. The Greek letter \(μ\) is the symbol for the population mean and \(\overline x\) is the symbol for the sample mean. Both formulas have a mathematical symbol that tells us how to make the calculations. It is called Sigma notation because the symbol is the Greek capital letter sigma: \(Σ\). Like all mathematical symbols it tells us what to do: just as the plus sign tells us to add and the \(x\) tells us to multiply. These are called mathematical operators. The \(Σ\) symbol tells us to add a specific list of numbers.

Let's say we have a sample of animals from the local animal shelter and we are interested in their average age. If we list each value, or observation, in a column, you can give each one an index number. The first number will be number 1 and the second number 2 and so on.

The mean (Arithmetic), median and mode are all measures of the "center" of the data, the "average". They are all in their own way trying to measure the "common" point within the data, that which is "normal". In the case of the arithmetic mean this is solved by finding the value from which all points are equal linear distances. We can imagine that all the data values are combined through addition and then distributed back to each data point in equal amounts. The sum of all the values is what is redistributed in equal amounts such that the total sum remains the same.

The geometric mean redistributes not the sum of the values but the product of multiplying all the individual values and then redistributing them in equal portions such that the total product remains the same. This can be seen from the formula for the geometric mean, \(\tilde x\): (Pronounced x-tilde)

\(\tilde{x}=\left(\prod_{i=1}^{n} x_{i}\right)^{\frac{1}{n}}=\sqrt[n]{x_{1} \cdot x_{2} \cdots x_{n}}=\left(x_{1} \cdot x_{2} \cdots x_{n}\right)^{\frac{1}{n}}\)

where \(π\) is another mathematical operator, that tells us to multiply all the \(x_i\) numbers in the same way capital Greek sigma tells us to add all the \(x_i\) numbers. Remember that a fractional exponent is calling for the nth root of the number thus an exponent of 1/3 is the cube root of the number.

The geometric mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same product?" The geometric mean gets its name from the fact that when redistributed in this way the sides form a geometric shape for which all sides have the same length. To see this, take the example of the numbers 10, 51.2 and 8. The geometric mean is the product of multiplying these three numbers together (4,096) and taking the cube root because there are three numbers among which this product is to be distributed. Thus the geometric mean of these three numbers is 16. This describes a cube 16x16x16 and has a volume of 4,096 units.

The geometric mean is relevant in Economics and Finance for dealing with growth: growth of markets, in investment, population and other variables the growth in which there is an interest. Imagine that our box of 4,096 units (perhaps dollars) is the value of an investment after three years and that the investment returns in percents were the three numbers in our example. The geometric mean will provide us with the answer to the question, what is the average rate of return: 16 percent. The arithmetic mean of these three numbers is 23.6 percent. The reason for this difference, 16 versus 23.6, is that the arithmetic mean is additive and thus does not account for the interest on the interest, compound interest, embedded in the investment growth process. The same issue arises when asking for the average rate of growth of a population or sales or market penetration, etc., knowing the annual rates of growth. The formula for the geometric mean rate of return, or any other growth rate, is:

\(r_s=(x_1  \cdot x_2⋅⋅⋅x_n)^{\frac{1}{n}}−1\)

Manipulating the formula for the geometric mean can also provide a calculation of the average rate of growth between two periods knowing only the initial value \(a_0\) and the ending value \(a_n\) and the number of periods, \(n\). The following formula provides this information:

\((\frac{a_n}{a_0})^{\frac{1}{n}}= \tilde x\)

Finally, we note that the formula for the geometric mean requires that all numbers be positive, greater than zero. The reason of course is that the root of a negative number is undefined for use outside of mathematical theory. There are ways to avoid this problem however. In the case of rates of return and other simple growth problems we can convert the negative values to meaningful positive equivalent values. Imagine that the annual returns for the past three years are +12%, -8%, and +2%. Using the decimal multiplier equivalents of 1.12, 0.92, and 1.02, allows us to compute a geometric mean of 1.0167. Subtracting 1 from this value gives the geometric mean of +1.67% as a net rate of population growth (or financial return). From this example we can see that the geometric mean provides us with this formula for calculating the geometric (mean) rate of return for a series of annual rates of return:

\(r_s = \tilde x−1\)

where \(r_s\) is average rate of return and \( \tilde x\) is the geometric mean of the returns during some number of time periods. Note that the length of each time period must be the same.

As a general rule one should convert the percent values to its decimal equivalent multiplier. It is important to recognize that when dealing with percents, the geometric mean of percent values does not equal the geometric mean of the decimal multiplier equivalents and it is the decimal multiplier equivalent geometric mean that is relevant.

Consider the following data set.
4; 5; 6; 6; 6; 7; 7; 7; 7; 7; 7; 8; 8; 8; 9; 10

This data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval.

Figure 2.11

The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.

The histogram for the data: 4; 5; 6; 6; 6; 7; 7; 7; 7; 8 shown in Figure 2.11 is not symmetrical. The right-hand side seems "chopped off" compared to the left side. A distribution of this type is called skewed to the left because it is pulled out to the left. We can formally measure the skewness of a distribution just as we can mathematically measure the center weight of the data or its general "speadness". The mathematical formula for skewness is: \(a_3=∑ \frac{(x_i−\overline x)^3}{ns^3}\). The greater the deviation from zero indicates a greater degree of skewness. If the skewness is negative then the distribution is skewed left as in Figure 2.12. A positive measure of skewness indicates right skewness such as Figure 2.13.

Figure 2.12

The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so.

The histogram for the data: 6; 7; 7; 7; 7; 8; 8; 8; 9; 10 shown in Figure 2.12, is also not symmetrical. It is skewed to the right.

Figure 2.13

The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most.

To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.

As with the mean, median and mode, and as we will see shortly, the variance, there are mathematical formulas that give us precise measures of these characteristics of the distribution of the data. Again looking at the formula for skewness we see that this is a relationship between the mean of the data and the individual observations cubed.

\(a_3=∑ \frac{(x_i − \overline x)^3}{ns^3}\)

where s is the sample standard deviation of the data, \(X_i\) , and \(\overline x\) is the arithmetic mean and \(n\) is the sample size.

Formally the arithmetic mean is known as the first moment of the distribution. The second moment we will see is the variance, and skewness is the third moment. The variance measures the squared differences of the data from the mean and skewness measures the cubed differences of the data from the mean. While a variance can never be a negative number, the measure of skewness can and this is how we determine if the data are skewed right of left. The skewness for a normal distribution is zero, and any symmetric data should have skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. The skewness characterizes the degree of asymmetry of a distribution around its mean. While the mean and standard deviation are dimensional quantities (this is why we will take the square root of the variance ) that is, have the same units as the measured quantities \(X_i\), the skewness is conventionally defined in such a way as to make it nondimensional. It is a pure number that characterizes only the shape of the distribution. A positive value of skewness signifies a distribution with an asymmetric tail extending out towards more positive \(X\) and a negative value signifies a distribution whose tail extends out towards more negative \(X\). A zero measure of skewness will indicate a symmetrical distribution.

Skewness and symmetry become important when we discuss probability distributions in later chapters.

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean.

The standard deviation

• provides a numerical measure of the overall amount of variation in a data set, and
• can be used to determine whether a particular data value is close to or far from the mean.
  

The standard deviation provides a measure of the overall variation in a data set
The standard deviation is always positive or zero. The standard deviation is small when the data are all concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.

Suppose that we are studying the amount of time customers wait in line at the checkout at supermarket A and supermarket B. The average wait time at both supermarkets is five minutes. At supermarket A, the standard deviation for the wait time is two minutes; at supermarket B. The standard deviation for the wait time is four minutes.

Because supermarket B has a higher standard deviation, we know that there is more variation in the wait times at supermarket B. Overall, wait times at supermarket B are more spread out from the average; wait times at supermarket A are more concentrated near the average.

Calculating the Standard Deviation

If \(x\) is a number, then the difference "\(x\) minus the mean" is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols a deviation is \(x – μ\). For sample data, in symbols a deviation is \(x – \overline x\).

The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lower case letter s represents the sample standard deviation and the Greek letter \(σ\) (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of \(σ\).

To calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the deviations (the \(x – \overline x \) values for a sample, or the \(x – μ\) values for a population). The symbol \(σ^2\) represents the population variance; the population standard deviation \(σ\) is the square root of the population variance. The symbol \(s^2\) represents the sample variance; the sample standard deviation \(s\) is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations. Formally, the variance is the second moment of the distribution or the first moment around the mean. Remember that the mean is the first moment of the distribution.

If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by \(N\), the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by \(n – 1\), one less than the number of items in the sample.

Formulas for the Sample Standard Deviation

• \(s=\sqrt{\frac{\Sigma(x-\bar{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\Sigma f(x-\bar{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\left(\sum_{i=1}^{n} x^{2}\right)-n \bar{x}^{2}}{n-1}}\)
  
• For the sample standard deviation, the denominator is n - 1, that is the sample size minus 1.

Formulas for the Population Standard Deviation

• \(\sigma=\sqrt{\frac{\Sigma(x-\mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\Sigma f(x-\mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\sum_{i=1}^{N} x_{i}^{2}}{N}-\mu^{2}}\)
  
  
• For the population standard deviation, the denominator is \(N\), the number of items in the population.
  

In these formulas, \(f\) represents the frequency with which a value appears. For example, if a value appears once, \(f\) is one. If a value appears three times in the data set or population, \(f\) is three. Two important observations concerning the variance and standard deviation: the deviations are measured from the mean and the deviations are squared. In principle, the deviations could be measured from any point, however, our interest is measurement from the center weight of the data, what is the "normal" or most usual value of the observation. Later we will be trying to measure the "unusualness" of an observation or a sample mean and thus we need a measure from the mean. The second observation is that the deviations are squared. This does two things, first it makes the deviations all positive and second it changes the units of measurement from that of the mean and the original observations. If the data are weights then the mean is measured in pounds, but the variance is measured in pounds-squared. One reason to use the standard deviation is to return to the original units of measurement by taking the square root of the variance. Further, when the deviations are squared it explodes their value. For example, a deviation of 10 from the mean when squared is 100, but a deviation of 100 from the mean is 10,000. What this does is place great weight on outliers when calculating the variance.

Types of Variability in Samples

When trying to study a population, a sample is often used, either for convenience or because it is not possible to access the entire population. Variability is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following:

• Observational or measurement variability
• Natural variability
• Induced variability
• Sample variability
  

Here are some examples to describe each type of variability.

Example 1: Measurement variability

Measurement variability occurs when there are differences in the instruments used to measure or in the people using those instruments. If we are gathering data on how long it takes for a ball to drop from a height by having students measure the time of the drop with a stopwatch, we may experience measurement variability if the two stopwatches used were made by different manufacturers: For example, one stopwatch measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The differences in outcomes may be affected by measurement variability.

Example 2: Natural variability

Natural variability arises from the differences that naturally occur because members of a population differ from each other. For example, if we have two identical corn plants and we expose both plants to the same amount of water and sunlight, they may still grow at different rates simply because they are two different corn plants. The difference in outcomes may be explained by natural variability.

Example 3: Induced variability

Induced variability is the counterpart to natural variability; this occurs because we have artificially induced an element of variation (that, by definition, was not present naturally): For example, we assign people to two different groups to study memory, and we induce a variable in one group by limiting the amount of sleep they get. The difference in outcomes may be affected by induced variability.

Example 4: Sample variability

Sample variability occurs when multiple random samples are taken from the same population. For example, if I conduct four surveys of 50 people randomly selected from a given population, the differences in outcomes may be affected by sample variability.

Example 2.29

In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of n = 20 fifth grade students. The ages are rounded to the nearest half year:

9; 9.5; 9.5; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5;

\(\overline x =\frac{9 + 9.5(2) + 10(4) + 10.5(4) + 11(6) + 11.5(3)}{20}=10.525\)

The average age is 10.53 years, rounded to two places.

The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating s.

The sample variance, \(s^2\), is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 – 1):

\(s_2=\frac{9.7375}{20−1}=0.5125\)

The sample standard deviation s is equal to the square root of the sample variance:

\(s=\sqrt{0.5125} = 0.715891\), which is rounded to two decimal places, \(s = 0.72\).

Explanation of the standard deviation calculation shown in the table

The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11 which is indicated by the deviations 0.97 and 0.47. A positive deviation occurs when the data value is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean. The deviation is –1.525 for the data value nine. If you add the deviations, the sum is always zero. (For Example 2.29, there are n = 20 deviations.) So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation. By squaring the deviations we are placing an extreme penalty on observations that are far from the mean; these observations get greater weight in the calculations of the variance. We will see later on that the variance (standard deviation) plays the critical role in determining our conclusions in inferential statistics. We can begin now by using the standard deviation as a measure of "unusualness." "How did you do on the test?" "Terrific! Two standard deviations above the mean." This, we will see, is an unusually good exam grade.

The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.

Notice that instead of dividing by \(n = 20\), the calculation divided by \(n – 1 = 20 – 1 = 19\) because the data is a sample. For the sample variance, we divide by the sample size minus one \((n – 1)\). Why not divide by \(n\)? The answer has to do with the population variance. The sample variance is an estimate of the population variance. This estimate requires us to use an estimate of the population mean rather than the actual population mean. Based on the theoretical mathematics that lies behind these calculations, dividing by \((n – 1)\) gives a better estimate of the population variance.

The standard deviation, \(s\) or \(σ\), is either zero or larger than zero. Describing the data with reference to the spread is called "variability". The variability in data depends upon the method by which the outcomes are obtained; for example, by measuring or by random sampling. When the standard deviation is zero, there is no spread; that is, the all the data values are equal to each other. The standard deviation is small when the data are all concentrated close to the mean, and is larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make \(s\) or \(σ\) very large.

Example 2.30

Use the following data (first exam scores) from Susan Dean's spring pre-calculus class:

33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100

a. Create a chart containing the data, frequencies, relative frequencies, and cumulative relative frequencies to three decimal places.

b. Calculate the following to one decimal place:

1. The sample mean
2. The sample standard deviation
3. The median
4. The first quartile
5. The third quartile
6. IQR

a. See Table 2.29

b.

1. The sample mean = 73.5
2. The sample standard deviation = 17.9
3. The median = 73
4. The first quartile = 61
5. The third quartile = 90
6. IQR = 90 – 61 = 29
   
   

• For each data value \(x\), calculate how many standard deviations away from its mean the value is.
• Use the formula: \(x\) = mean + (#ofSTDEVs)(standard deviation); solve for #ofSTDEVs.
• #ofSTDEVs=\(\frac{x – mean}{standard \: deviation}\)
• Compare the results of this calculation.

Example 2.32

• At least 75% of the data is within two standard deviations of the mean.
• At least 89% of the data is within three standard deviations of the mean.
• At least 95% of the data is within 4.5 standard deviations of the mean.
• This is known as Chebyshev's Rule.

• Approximately 68% of the data is within one standard deviation of the mean.
• Approximately 95% of the data is within two standard deviations of the mean.
• More than 99% of the data is within three standard deviations of the mean.
• This is known as the Empirical Rule.
• It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.

Coefficient of Variation

Measures of Central Tendency

Mode, Median, Mean, and Midrange

This section explores the measures of central tendency: mode, median, mean and midrange. It shows you the commands for computing these metrics using a spreadsheet program and it gives you the correct format for entering those commands into the spreadsheet program.

Mode

The mode is the value that occurs most frequently in the data. Spreadsheet programs such as Microsoft Excel or OpenOffice.org Calc can determine the mode with the function MODE.

=MODE(data)

In the Fall of 2000 the statistics class gathered data on the number of siblings for each member of the class. One student was an only child and had no siblings. One student had 13 brothers and sisters. The complete data set is as follows:

1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 7, 8, 9, 10 ,12 ,12 ,13

The mode is 2 because 2 occurs more often than any other value. Where there is a tie there is no mode.

For the ages of students in that class

18, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 25, 25, 26

...there is no mode: there is a tie between 21 and 22, hence there no single must frequent value. Spreadsheets will, however, usually report a mode of 21 in this case. Spreadsheets often select the first mode in a multi-modal tie.

If all values appear only once, then there is no mode. Spreadsheets will display #N/A or #VALUE to indicate an error has occurred - there is no mode. No mode is NOT the same as a mode of zero. A mode of zero means that zero is the most frequent data value. Do not put the number 0 (zero) for "no mode." An example of a mode of zero might be the number of children for students in statistics class.

Median

The median is the central (or middle) value in a data set. If a number sits at the middle, then it is the median. If the middle is between two numbers, then the median is half way between the two middle numbers.

For the sibling data...

1, 2, 2, 2, 2, 2, 3, 3, 4, 4, |4| ,5 ,5 ,5 ,7 ,8 ,9 ,10 ,12 ,12 ,13

...the median is 4.

Note the data must be in order (sorted) before you can find the median. For the data 2, 4, 6, 8 the median is 5: (4+6)/2.

The median function in spreadsheets is MEDIAN.

=MEDIAN(data)

Mean (average)

The mean, also called the arithmetic mean and also called the average, is calculated mathematically by adding the values and then dividing by the number of values (the sample size n).

If the mean is the mean of a population, then it is called the population mean μ. The letter μ is a Greek lower case "m" and is pronounced "mu."

If the mean is the mean of a sample, then it is the sample mean \(\overline x\). The symbol \(\overline x\) is pronounced "x bar."

population mean \(μ = \frac{sum \: of \: the \: population \: data}{population \: size \:  N} = \frac{∑X}{N}\)

sample mean \(\overline x = \frac{sum \: of \: the \: sample\: data}{sample\: size \: N} = \frac{∑X}{N} \)

The sum of the data ∑ x can be determined using the function =SUM(data). The sample size n can be determined using =COUNT(data). Thus =SUM(data)/COUNT(data) will calculate the mean. There is also a single function that calculates the mean. The function that directly calculates the mean is AVERAGE

=AVERAGE(data)

Resistant measures: One that is not influenced by extremely high or extremely low data values. The median tends to be more resistant than mean.

Population mean and sample mean

If the mean is measured using the whole population then this would be the population mean. If the mean was calculated from a sample then the mean is the sample mean. Mathematically there is no difference in the way the population and sample mean are calculated.

Midrange

The midrange is the midway point between the minimum and the maximum in a set of data.

To calculate the minimum and maximum values, spreadsheets use the minimum value function MIN and maximum value function MAX.

=MIN(data)

=MAX(data)

The MIN and MAX function can take a list of comma separated numbers or a range of cells in a spreadsheet. If the data is in cells A2 to A42, then the minimum and maximum can be found from:

=MIN(A2:A42)

=MAX(A2:A42)

The midrange can then be calculated from:

midrange = (maximum + minimum)/2

In a spreadsheet use the following formula:

=(MAX(data)+MIN(data))/2

Source: Dana Lee Ling, http://www.comfsm.fm/~dleeling/statistics/text5.html