# Statistics

## Gathering and Organizing Data


### Learning Objectives

After completing this section, you should be able to:
1. Distinguish among sampling techniques.
2. Organize data using an appropriate method.
3. Create frequency distributions.
When a polling organization wants to try to establish which candidate will win an upcoming election, the first steps are to write questions for the survey and to choose which people will be asked to respond to the survey. These can seem like simple steps, but they have far-reaching implications in the analysis the pollsters will later carry out. The process by which samples (or groups of units from which we collect data) are chosen can strongly affect the data that are collected. Units are anything that can be measured or surveyed (such as people, animals, objectives, or experiments) and data are observations made on units.
One of the most famous failures of good sampling occurred in the first half of the twentieth century. The Literary Digest was among the most respected magazines of the early twentieth century. Despite the name, the Digest was a weekly newsmagazine. Starting in 1916, the Digest conducted a poll to try to predict the winner of each US Presidential election. For the most part, their results were good; they correctly predicted the outcome of all five elections between 1916 and 1932. In 1936, the incumbent President Franklin Delano Roosevelt faced Kansas governor Alf Landon, and once again the Digest ran their famous poll, with results published the week before the election. Their conclusion? Landon would win in a landslide, 57% to 43%. Once the actual votes had been counted, though, Roosevelt ended up with 61% of the popular vote, 18% more than the poll predicted. What went wrong?
The short answer is that the people who were chosen to receive the survey (over ten million of them!) were not a good representation of the population of voting adults. The sample was chosen using the Digest's own base of subscribers as well as publicly available lists of people that were likely adults (and therefore eligible to vote), mostly phone books and vehicle registration records. The pollsters then mailed every single person on these lists a survey. Around a quarter of those surveys were returned; this constituted the sample that was used to make the Digest’s disastrously incorrect prediction. However, the Digest made an error in failing to consider that the election was happening during the Great Depression, and only the wealthy had disposable income to spend on telephone lines, automobiles, and magazine subscriptions. Thus, only the wealthy were sent the Digest’s survey. Since Roosevelt was extremely popular among poorer voters, many of Roosevelt’s supporters were excluded from the Digest’s sample.
Another more complicated factor was the low response rate; only around 25% of the surveys were returned. This created what’s called a non-response bias.

### Sampling and Gathering Data

The Digest's failure highlights the need for what is now considered the most important criterion for sampling: randomness. This randomness can be achieved in several ways. Here we cover some of the most common.
A simple random sample is chosen in a way that every unit in the population has an equal chance of being selected, and the chances of a unit being selected do not depend on the units already chosen. An example of this is choosing a group of people by drawing names out of a hat (assuming the names are well-mixed in the hat).
A systematic random sample is selected from an ordered list of the population (for example, names sorted alphabetically or students listed by student ID). First, we decide what proportion of the population will be in our sample. We want to express that proportion as a fraction with 1 in the numerator. Let’s call that number D. Next, we’ll choose a random number between one and D. The unit at that position will go into our sample. We’ll find the rest of our sample by choosing every Dth unit in the list, starting with our random number.
To walk through an example, let’s say we want to sample 2% of the population: . (Note: If the number in the denominator isn’t a whole number, we can just round it off. This part of the process doesn’t have to be precise.) We can then use a random number generator to find a random number between 1 and 50; let's use 31. In our example, our sample would then be the units in the list at positions 31, 81 (31 + 50), 131 (81 + 50), and so forth.
A stratified sample is one chosen so that particular groups in the population are certain to be represented. Let’s say you are studying the population of students in a large high school (where the grades run from 9th to 12th), and you want to choose a sample of 12 students. If you use a simple or systematic random sample, there’s a pretty good chance that you’ll miss one grade completely. In a stratified sample, you would first divide the population into groups (the strata), then take a random sample within each stratum (that’s the singular form of “strata”). In the high school example, we could divide the population into grades, then take a random sample of three students within each grade. That would get us to the 12 students we need while ensuring coverage of each grade.
A cluster sample is a sample where clusters of units are chosen at random, instead of choosing individual units. For example, if we need a sample of college students, we may take a list of all the course sections being offered at the college, choose three of them at random (the sections are the clusters), and then survey all the students in those sections. A sample like this one has the advantage of convenience: If the survey needs to be administered in person, many of your sample units will be located in one place at the same time.

### Organizing Data

Once data have been collected, we turn our attention to analysis. Before we analyze, though, it’s useful to reorganize the data into a format that makes the analysis easier. For example, if our data were collected using a paper survey, our raw data are all broken down by respondent (represented by an individual response sheet). To perform an analysis on all the responses to an individual question, we need to first group all the responses to each question together. The way we organize the data depends on the type of data we’ve collected.
There are two broad types of data: categorical and quantitative.  Categorical data classifies the unit into a group (or category). Examples of categorical data include a response to a yes-or-no question, or the color of a person’s eyes. Quantitative data is a numerical measure of a property of a unit. Examples of quantitative data include the time it takes for a rat to run through a maze or a person’s daily calorie intake. We’ll look at each type of data in turn when considering how best to organize.

### Categorical Data Organization

The best way to organize categorical data is using a categorical frequency distribution. A categorical frequency distribution is a table with two columns. The first contains all the categories present in the data, each listed once. The second contains the frequencies of each category, which are just a count of how often each category appears in the data.

### Quantitative Data

We have a couple of options available for organizing quantitative data. If there are just a few possible responses, we can create a frequency distribution just like the ones we made for categorical data above. For example, if we’re surveying a group of high school students and we ask for each student’s age, we’ll likely only get whole-number responses between 13 and 19. Since there are only around seven (and likely fewer) possible responses, we can treat the data as if they’re categorical and create a frequency distribution as before.
If there are many possible responses, a frequency distribution table like the ones we’ve seen so far isn’t really useful; there will likely be many responses with a frequency of one, which means the table will be no better than looking at the raw data. In these cases, we can create a binned frequency distribution. A binned frequency distribution groups the data into ranges of values called bins, then records the number of responses in each bin.
For example, if we have height data for individuals measured in centimeters, we might create bins like 150–155 cm, 155–160 cm, and so forth (making sure that every data value falls into a bin). We must be careful, though; in this scenario, it’s not clear which bin would contain a response of 155 cm. Usually, responses on the edge of a bin are placed in the higher bin, but it’s good practice to make that clear. In cases where responses are rounded off, you can avoid this issue by leaving a gap between the bins that couldn’t contain any responses. In our example, if the measurements were all rounded off to the nearest centimeter, we could make bins like 150–154 cm, 155–159 cm, etc. (since a response like 154.2 isn’t possible). We’ll use this method going forward. How do we decide what the boundaries of our bins should be? There’s no one right way to do that, but there are some guidelines that can be helpful.
1. Every data value should fall into exactly one bin. For example, if the lowest value in our data is 42, the lowest bin should not be 45–49.
2. Every bin should have the same width. Note that if we shift the upper limits of our bins down a bit to avoid ambiguity (like described above), we can’t simply subtract the lower limit from the upper limit to get the bin width; instead, we subtract the lower limit of the bin from the lower limit of the next bin. For example, if we’re looking at GPAs rounded to the nearest hundredth, we might choose bins like 2.00–2.24, 2.25–2.49, 2.50–2.74, etc. These bins all have a width of 0.25.
3. If the minimum or maximum value of the data falls right on the boundary between two bins, then it’s OK to bend the rule just a little in order to avoid having an additional bin containing just that one value. We’ll see an example of this in just a moment.
4. If we have too many or too few bins, it can be difficult to get a good sense of the distribution. Seven or eight bins is ideal, but that’s not a firm rule; anything between five and twelve is fine. We often choose the number of bins so that the widths are round numbers.

### Check Your Understanding


### Key Terms

1. sample
2. units
3. data
4. population
5. simple random sample
6. systematic random sample
7. stratified sample
8. cluster sample
9. quantitative data
10. categorical data
11. categorical frequency Distribution
12. binned frequency distribution

### Key Concepts

1. Categorical data places units into groups (categories), while quantitative data is a numerical measure of a property of a unit.
2. The sampling method for a study depends on the way that randomization is used to select units for the sample.
3. Frequency distributions help to summarize data by counting the number of units that fall into a particular category or range of quantitative values.