MA121: Introduction to Statistics

1a. Describe various types of sampling methods to data collection, and apply these methods

• Explain the difference between descriptive and inferential statistics.
• What is the difference between quantitative and qualitative data?
• Define and describe the difference between the two types of quantitative data, discrete and continuous.
• Define bias and inferential data.
• How can researchers control for possible bias in samples?
• Define sample and sample size.
• Define three types of sampling: stratified, cluster and systematic.

Descriptive statistics provide facts about a set of data, which are often depicted in a graph. Where is the data centered? What is the mean? What is the median? Are the data centered or bell-shaped? Are the data skewed to the right or left. In other words do most of the values appear at different ends of the graph? These descriptions tell us facts about the data in front of us and about the entire population or sample we have chosen.

Inferential statistics refers to how we draw conclusions about a population of data, when we examine data in a sample. Inferential statistics is the more common method of statistics, since we rarely have the time or resources to survey or measure every item, member, or person in a given group or population. In statistics, differences among various types of data is important since they determine which types of tests or graphs are most useful for displaying data and answering certain questions.

Quantitative (or numeric) data describes data that are numeric or mathematical. We use quantitative data to calculate sums, averages, means, other types of statistics, and other mathematical operations.

Qualitative (or categorical) data describes data that are non-numeric. Qualitative data includes text, letters, and words. Qualitative data can also include numerical digits, but mathematical operations do not make sense or may be impossible in this usage.

For example, think of a phone number, zip code, or postal code. Although these designations consist of numerical digits (numbers) and may include dashes and parentheses, we do not use them to conduct calculations. We do not add up or calculate an average for the numbers in our telephone contact list or postal code. Phone numbers, zip codes, and postal codes are examples of qualitative data.

We can categorize quantitative data further as discrete or continuous data.

A discrete data set contains a fixed, or small, number of possible values. The classic example of a discrete data set is a six-sided die. You can only roll a one, two, three, four, five or six. You cannot roll a 2.8.

A continuous data set has a large or infinite number of possibilities. Consider the weight of a group of college students. When we discard extreme outliers, our range may be from 120 to 200 pounds. This dataset includes 61 possible values, assuming we measure to the whole pound. When a 285-pound football player enters our dataset, we must add an additional 85 possibilities, to account for 166 possible values, because the dataset is continuous. Although this number is not infinite, it is impractical to consider it discrete data.

Bias occurs when we deal with inferential data. For example, because we cannot practically consider every person in the United States as part of our dataset, we have to make sure our sample is representative of the entire population.

A common example of how things can go wrong is the 1936 U.S. presidential election when political polling was brand new. The Literary Digest magazine mailed thousands of survey cards to its readers and people it found in the phone book and car registration directory to poll whether Republican challenger Alf Landon would defeat the Democrat presidential incumbent Franklin D. Roosevelt. The responses predicted Landon would win in a landslide. What went wrong?

Their sample size was large enough and their mathematical calculations were probably correct. However, in 1936 phones, cars, and even magazine subscriptions, were considered luxury items. The magazine's sample disproportionately represented wealthier Americans who were more likely to vote Republican.

While it is difficult to eliminate sampling bias entirely, we can use different sampling methods to reduce it.

1. Stratified sampling divides the population into sub-populations and takes a random sample of each group. For example, if you think Democrats, Republicans, and Independents are going to poll differently on an issue, and these groups each represent 30, 25, and 45 percent of the entire U.S. population, your sample should reflect the same proportion of representatives. You should choose 30 Democrats, 25 Republicans, and 45 independents at random and send the survey form to all of these 100 individuals.
   
   
2. Researchers use cluster sampling when their population is already divided into representative groups. For example, a researcher might study buildings (or clusters) in an apartment complex, choose a certain number of buildings at random, and sample everyone in each building.
   
   
3. Researchers use systematic sampling when they have a rough idea of population size, but lack a representative cluster to sample. For example, a researcher might poll every tenth person who walks through the door in a shopping mall or an inspector might examine every 20th item on an assembly line for quality.

Review this material in Basic Definitions and Concepts, Descriptive Statistics, Inferential Statistics, and Variables.

1b. Create and interpret frequency tables

• What is a frequency table?
• Define class, bin, and class intervals.
• Define outlier.
• Why is it important to include every possible value between the lowest and highest in the left side of a frequency table, even when that value is not represented or has no frequency? In other words, if the data you obtain from an experiment ranges from 1 to 5 and there are no 4s, why do you need to display a row for 4?
  
• What do you do when there are too many values in a data set to give each variable its own row? Let's say the variable is yards rushing during a football game and the possible values are 51 to 218. Your space does not allow you to display more than 160 rows in your table. What do you do?

A frequency table lists every possible value of the random variable (every possible value) in a data distribution. It must include an interior value, even when that data point has no frequency. This is because one of the purposes of a frequency table is to display how varied the data is.

For example, if our values are 1, 2, 3, and 25, you should space the 25 so it is 22 points away from the three to illustrate how much of an outlier it is. This means you should include rows for 4 through 24, with a zero frequency for each.

If the distribution has too many possible values, you can group the values into class intervals (sometimes called classes or bins) and mark the frequency for each.

Let's return to our football example. You can display the number of players who have rushed 50 to 59 yards, 60 to 69 yards, and so on. If you treat this data as discrete, it will be a long, tedious table with mostly ones and zeros, since you would need to show the number of players who rushed 50 yards, the number of players who rushed 51 yards, the number of players who rushed 52 yards, and so on. Grouping or organizing the figures into classes of ten yards each is more concise and gives the reader a more comprehensive picture of the total data.

If you group your data into classes make sure:

1. Each class is the same width, 
2. Each class does not overlap, and 
3. Each class accounts for every possible value.

For example, we could use 50 to 59 yards, 60 to 69 yards, and so on. Each class width would equal ten rushing yards. We should not use 50 to 55 yards, 56 to 70 yards, or 71 to 73 yards.

Note that we are assuming the measurements are in whole yards. If we had a measurement of 55.3 in the above grouping, we would violate our third rule (that each class accounts for every possible value) since 55.3 comes in between two classes. So you may have to group your data as 50 to 55, 55 to 60, and so on, so you would put 55.5, and even 55.99 in the first class or bin.

Your decision on how wide to make your class interval and how many intervals to include, is really a matter of personal preference. The course textbook refers to formulas like Sturges' Rule and Rice Rule which are easy to compute (the number of intervals equals the cube root of the number of observations). But you may need to make some adjustments since your result will probably not be a whole number (integer).

Do not worry too much about memorizing these rules, since statisticians do not even agree on which rule is best. Researchers generally recommend using 5 to 20 class widths, but this can vary depending on whether your data is homogeneous or heterogeneous. It might be a good strategy to put your data in groups of tens so you can classify each data point by its first digit.

Basically, if you have too few classes, your readers may not appreciate the diversity of your data. If you have too many classes, you will probably display a lot of ones and zeros which can become tedious for the reader. Experiment with a few options until you see one that makes sense to you and your audience.

Review this material in Histograms and Frequency Distribution.

1c. Display data graphically and interpret the following types of graphs: stem plots, histograms, and boxplots

• Define a histogram. How does it differ from a bar graph? What are some rules for drawing histograms that do not apply to bar graphs?
• How is a stem plot similar to and different from a histogram?
• What is a dot plot?
• What does a box plot represent graphically?

A histogram is a special type of bar graph researchers use to display quantitative distributions. A histogram differs from a bar graph because the horizontal axis is numeric: the horizontal axis of a bar graph represents qualitative data.

A histogram should follow the same three rules for frequency tables listed above. The numbers on the horizontal axis need to be in order if they are grouped into classes. This makes it easy for readers to recognize any data point that is an outlier, due to the distance of the outlier's bar from the other bars in the graph. You must include any data point or interval that has zero values, with no bar. It is also important that the bars do not touch. For consistency, data that is homogeneous (or heterogeneous) is organized in the same way on the histogram. Remember that a histogram's main purpose is to represent a frequency table graphically.

A stem plot (also called a stem-and-leaf plot) is similar to a histogram, except it includes the last digit of the actual data values (the leaves) above the stem of the first digit or digits. Researchers use a stem plot to display the actual data on their graph. Stem plots quickly convey the minimum, maximum, and median of the data points to their readers.

A stem plot is similar to a dot plot which uses dots or similar markings to represent data points. For example, if the bar height (frequency) of the histogram is seven, and the values are 50, 51, 52, 54, 55, 56, 59, the histogram will display a bar that is seven units high. A dot plot will have seven dots going up from the horizontal axis, and a stem plot will have "5" in the stem, with the digits 1, 2, 4, 5, 6, 9 written out to the right.

A box plot graphically displays the five-number-summary of a data set. The five numbers are: minimum, first quartile, median, third quartile, and maximum. We will review the five-number summary in later units, but think of the first and third quartiles as the median of the lower and upper halves of the distribution, respectively. In other words, the five numbers partition the data set into four quartiles, each with (roughly) the same number of data points.

Review this material in Histograms, Stem and Leaf Displays, Dot Plots, and Box Plots.

1d. Identify, describe, and calculate the following measures of the location of data: quartiles and percentiles

• What is a percentile, and how is it related to or different from a quartile?
• What is the relationship between quartiles and the median?
• What is the five-number summary?
• My calculator and Excel give me two different numbers for the 1st and 3rd quartile. Why is that?

The Pth percentile of a data set means that P% of the data falls below that number and (100−P)% fall above that number. For example, the 80th percentile is the number for which 80% of the data is below and 20% is above. The median (half the data below, half above) is also the 50th percentile. These are approximate, especially for small data sets. This is because, for example, if you have 20 data points, finding the 37th percentile is not going to be an exact number. Since there are only 20 numbers (assuming no repeats), you'll have a 35th percentile and the next number up will be the 40th.

The quartiles of a data set divide that data set into four roughly-equally-sized (number of data points) parts. Again, we say "roughly" because if the number of data points is somewhat small and is not a multiple of 4 (17 data points, for example), you are not going to get four quartiles of the same size. This is another reason why, as we state above, finding quartiles is approximate.

When you have very large data sets (100 or more), you can find theoretical percentiles by using the Normal Distribution (see Unit 2). The median is the same thing as the 2nd quartile. The 1st quartile divides the first half of the data in half, and the 3rd quartile splits the upper half of the data. The minimum, 1st quartile, median/2nd quartile, 3rd quartile, and maximum make up the boundaries of the four quarters of data, and are referred to as the five number summary.

Finding quartiles for small data sets can be tricky, and even a bit subjective, if the number of data points is not a multiple of four. In the set: {0, 3, 4, 6, 8, 11, 11, 13, 15, 17, 20, 25}, the median is 11, the first quartile (separating the 3rd and 4th point) is 5, the third quartile (separating the 9th and 10th point) is 16. This is relatively simple because we have 12 data points and we can evenly divide them into groups of 3: {0, 3, 4 || 6, 8, 11 || 11, 13, 15 || 17, 20, 25}.

However, let's say we have 15 data points: {1, 3, 4, 5, 7, 8, 10, 10, 11, 13, 15, 16, 19, 20, 25}. The 2Q/median is the 8th data point (10). What about the 1st quartile? If we include the median, the first half is {1, 3, 4, 5, 7, 8, 10, 10}, in which case the 1st quartile (the median of this set) is 6. If we exclude the median {1, 3, 4, 5, 7, 8, 10}, then the 1st quartile is 5. That is why different technologies may give you different answers. Which is correct? Well, both. Even Microsoft Excel has two different functions for quartiles, one inclusive and one exclusive. This discrepancy disappears as the data set gets very large. 

Review this material in Percentiles and Relative Position of Data.

1e. Identify, describe, and calculate the measures of the center of mean, median, and mode

• Explain the differences between mean and median.
• What does the center of distribution tell us?
• When is the median a better measure of center than the mean?
• When is mode preferable to the mean or median?

We calculate the mean by adding all of the data points and dividing the result by the size. The mean provides a rough idea of the center of the distribution. The median does this too, but disregards all data points except for the one or ones in the middle.

You can think of the median as more resistant to outliers. In other words, when a researcher adds a significantly higher or lower value to a data set, or if the data is right or left skewed, the mean will adjust accordingly, with a significant upward or downward effect. The median, on the other hand, disregards the high and low values, so adding extreme values will have a much smaller, if any, effect on the median.

The mode conveys the most common data type. It is different from the mean and median because it is the only measure of center we can use with qualitative data, since it does not require a calculation or computation.

Review this material in Mean, Median, and Mode; Variance, Measures of Central Tendency, Median and Mean, and Mean and Median Demonstration.

1f. Identify, describe, and calculate the following measures of the spread of data: variance, standard deviation, and range

• Why is range generally not a reliable measure of spread?
• Why are measures of spread necessary? What critical information does the measure of center fail to provide?
• What is the difference between variance and standard deviation?

Measures of spread are just as important as measures of center. The mean and median, for quantitative data, give us an idea about where the center of the distribution is.

The variance and standard deviation tell us about the spread of the data, or how varied or heterogeneous your data are. A variance of zero happens if all data points are equal. For example, the data sets {49, 50, 51} and {0, 50, 100} have the same mean and median (50), but the second set is much more varied. We quantify this variability with the measure of spread.

The variance equals the mean of the squared differences between each data point and the mean.

The standard deviation equals the square root of the variance. One of the reasons we compute this is to get the units back to the original data set. If the data points are in minutes, the unit for the variance would be "square minutes" which does not make sense.

The simplest measure to use is the range (maximum to minimum), but the range is generally not a good measure since it only takes the highest and lowest data points into account.

Consider Data Set A = {47, 48, 49, 50, 51} and Data Set B = {47, 48, 49, 50, 51, 100}.

The range goes from four to 53. Set B is certainly more varied, but the 100 is an outlier, so the change in standard deviation is less extreme. The standard deviation is 1.6 for set A and 20.9 for set B.

Review this material in Measures of Variability and Mean, Median, and Mode; Variance.

Unit 1 Vocabulary

• Bar graph
• Bell-shaped
• Bias
• Bins
• Box plot
• Centered
• Classes
• Class intervals
• Cluster sampling
• Continuous data
• Descriptive statistics
• Discrete data
• Dot plot
• First quartile
• Five-number summary
• Frequency table
• Histogram
• Horizontal axis
• Inferential statistics
• Leaf
• Mean
• Measure of center
• Median
• Mode
• Outlier
• Qualitative data
• Quantitative data
• Quartiles
• Range
• Representative
• Resistant
• Sample
• Sampling method
• Skewed left
• Skewed right
• Spread
• Standard deviation
• Stem
• Stem plot
• Stem and leaf plot
• Stratified sampling
• Systematic sampling
• Third quartile
• Variance
• Variation