Chapter 15

Chapter 15
Descriptive Statistics

An overview of the field of statistics is shown in Figure 15.1 (also shown below). As you can see, the field of statistics can be divided into descriptive statistics and inferential statistics (and there are further subdivisions under inferential statistics which is the topic of the next chapter).

This chapter is about descriptive statistics (i.e., the use of statistics to describe, summarize, and explain or make sense of a given set of data).

A data set (i.e., a set of data with the "cases" going down the rows and the "variables" going across the columns) is shown in Table 15.1.
Once you put your data set (such as the one in Table 15.1) into a statistical program such as SPSS, you are ready to obtain all the descriptive statistics that you want (i.e., which will help you to make some sense out of your data).

Frequency Distributions

One useful way to view the data of a variable is to construct a frequency distribution (i.e., an arrangement in which the frequencies, and sometimes percentages, of the occurrence of each unique data value are shown).

An example is shown in Table 15.2 in the book and here for your convenience.

When a variable has a wide range of values, you may prefer using a grouped frequency distribution (i.e., where the data values are grouped into intervals and the frequencies of the intervals are shown).

For the above frequency distribution, one possible set of grouped intervals would be 20,000-24,999; 25,000-29,999; 30,000-34,999; 35,000-39,999; 40,000-44,999.

Note that the categories developed for a grouped frequency distribution must be mutually exclusive (the property that intervals do not overlap) and exhaustive (the property that a set of intervals or categories covers the complete range of data values).

An example of a grouped frequency distribution is shown on pate 437.

Graphic Representations of Data

Another excellent way to describe your data (especially for visually oriented learners) is to construct graphical representations of the data (i.e., pictorial representations of the data in two-dimensional space).

Some common graphical representations are bar graphs, histograms, line graphs, and scatterplots.

Bar Graphs

A bar graph uses vertical bars to represent the data.

The height of the bars usually represent the frequencies for the categories that sit on the X axis.
Note that, by tradition, the X axis is the horizontal axis and the Y axis is the vertical axis.
Bar graphs are typically used for categorical variables.
Here is a bar graph of one of the categorical variables included in the data set for this chapter (i.e., the data set shown on page 435).

Histograms

A histogram is a graphic that shows the frequencies and shape that characterize a quantitative variable.

In statistics, we often want to see the shape of the distribution of quantitative variables; having your computer program provide you with a histogram is a simple way to do this.
Here is a histogram for a quantitative variable included in the data set for this chapter:

Line Graphs

A line graph uses one or more lines to depict information about one or more variables.

A simple line graph might be used to show a trend over time (e.g., with the years on the X axis and the population sizes on the Y axis).
Here is an example of a line graph (Figure 15.4):

Line graphs are used for many different purposes in research. For example, if you will turn to page 290, you will see a line graph (e.g., Figure 9.15) used in factorial experimental designs to depict the relationship between two categorical independent variables and the dependent variable.
Yet another line graph is shown on page 468 in the next chapter. This line graph shows that the "sampling distribution of the mean" is normally distributed.
As you can see in the Figures just listed, line graphs have in common their use of one or more lines within the graph (to depict the levels or characteristics of a variable or to depict the relationships among variables).

Scatterplots

A scatterplot is used to depict the relationship between two quantitative variables.

Typically, the independent or predictor variable is represented by the X axis (i.e., on the horizontal axis) and the dependent variable is represented by the Y axis (i.e., on the vertical axis).
Here is an example of a scatterplot showing the relationship between two of the quantitative variables from the data set for this chapter:

Measures of Central Tendency

Measures of central tendency provide descriptive information about the single numerical value that is considered to be the most typical of the values of a quantitative variable.

Three common measures of central tendency are the mode, the median, and the mean.

The mode is simply the most frequently occurring number.

The median is the center point in a set of numbers; it is also the fiftieth percentile.

To get the median by hand, you first put your numbers in ascending or descending order.
Then you check to see which of the following two rules applies:

· Rule One. If you have an odd number of numbers, the median is the center number (e.g., three is the median for the numbers 1, 1, 3, 4, 9).

· Rule Two. If you have an even number of numbers, the median is the average of the two innermost numbers (e.g., 2.5 is the median for the numbers 1, 2, 3, 7).

The mean is the arithmetic average (e.g., the average of the numbers 2, 3, 3, and 4, is equal to 3).

A Comparison of the Mean, Median, and Mode

The mean, median, and mode are affected by what is called skewness (i.e., lack of symmetry) in the data.

Here is Figure 15.6, which showed a normal curve, a negatively skewed curve, and a positively skewed curve:

Look at the above figure and note that when a variable is normally distributed, the mean, median, and mode are the same number.
When the variable is skewed to the left (i.e., negatively skewed), the mean shifts to the left the most, the median shifts to the left the second most, and the mode the least affected by the presence of skew in the data.
Therefore, when the data are negatively skewed, this happens:

mean < median < mode.

When the variable is skewed to the right (i.e., positively skewed), the mean is shifted to the right the most, the median is shifted to the right the second most, and the mode the least affected.
Therefore, when the data are positively skewed, this happens:

mean > median > mode.

If you go to the end of the curve, to where it is pulled out the most, you will see that the order goes mean, median, and mode as you “walk up the curve” for negatively and positively skewed curves.

You can use the following two rules to provide some information about skewness even when you cannot see a line graph of the data (i.e., all you need is the mean and the median):

1. Rule One. If the mean is less than the median, the data are skewed to the left.

2. Rule Two. If the mean is greater than the median, the data are skewed to the right.

Measures of Variability

Measures of variability tell you how "spread out" or how much variability is present in a set of numbers. They tell you how different your numbers tend to be. Note that measures of variability should be reported along with measures of central tendency because they provide very different but complementary and important information. To fully interpret one (e.g., a mean), it is helpful to know about the other (e.g., a standard deviation).

An easy way to get the idea of variability is to look at two sets of data, one that is highly variable and one that is not very variable.

For example, which of these two sets of numbers appears to be the most spread out, Set A or Set B?

Set A. 93, 96, 98, 99, 99, 99, 100
Set B. 10, 29, 52, 69, 87, 92, 100

If you said Set B is more spread out, then you are right! The numbers in set B are more "spread out"; that is, they are more variability.

All of the measures of variability should give us an indication of the amount of variability in a set of data. We will discuss three indices of variability: the range, the variance, and the standard deviation.

Range

A relatively crude indicator of variability is the range (i.e., which is the difference between the highest and lowest numbers).

For example the range in Set A shown above is 7, and the range in Set B shown above is 90.

Variance and Standard Deviation

Two commonly used indicators of variability are the variance and the standard deviation.

Higher values for both of these indicators indicate a larger amount of variability than do lower numbers.
Zero stands for no variability at all (e.g., for the data 3, 3, 3, 3, 3, 3, the variance and standard deviation will equal zero).
When you have no variability, the numbers are a constant (i.e., the same number).

Table 15.4 shows you how to easily calculate, by hand, the variance and standard deviation.

(Basically, you set up the three columns shown, get the sum of the third column, and then plug the relevant numbers into the variance formula.)
The variance tells you (exactly) the average deviation from the mean, in "squared units."
The standard deviation is just the square root of the variance (i.e., it brings the "squared units" back to regular units).
The standard deviation tells you (approximately) how far the numbers tend to vary from the mean. (If the standard deviation is 7, then the numbers tend to be about 7 units from the mean. If the standard deviation is 1500, then the numbers tend to be about 1500 units from the mean.)

Virtually everyone in education is already familiar with the normal curve (a picture of one is shown in Figure 15.7 on page 449).

If data are normally distributed, then an easy rule to apply to the data is what we call “the 68, 95, 99.7 percent rule." That is . . .

Approximately 68% of the cases will fall within one standard deviation of the mean.
Approximately 95% of the cases will fall within two standard deviations of the mean.
Approximately 99.7% of the cases will fall within three standard deviations of the mean.

Measures of Relative Standing

Measures of relative standing are used to provide information about where a particular score falls in relation to the other scores in a distribution of data. Two commonly used measures of relative standing are percentile ranks and Z-scores.

Here is Figure 15.8 which shows these and some additional types of standard scores. You can determine the mean of the type of standard scores below by simply looking under Mean. You can determine the standard deviation by looking at how much the scores increase as you move from the mean to 1 SD.

Z-Scores: have a mean of 0 and a standard deviation of 1. Therefore, if you converted any set of scores (e.g., the set of student grades on a test) to z-scores, then that new set WILL have a mean of zero and a standard deviation of one.
IQ has a mean of 100 and a standard deviation of 15.
SAT has a mean of 500 and a standard deviation of 100.
Note: percentile ranks are a different type of score; because they only have ordinal measurement properties, the concept of standard deviation is not relevant.

Percentile Ranks

A percentile rank tells you the percentage of scores in a reference group (i.e., in the norming group) that fall below a particular raw score.

For example, if your percentile rank is 93 then you know that 93 percent of the scores in the reference group fall below your score.

Z-Scores

A z-score tells you how many standard deviations (SD) a raw score falls from the mean.

A SD of 2 says a score falls two standard deviations above the mean.
A SD of -3.5 says the score falls three and a half standard deviations below the mean.

To transform a raw score into z-score units, just use the following formula:

                     Raw score - Mean
Z-score =     ------------------------
                    Standard Deviation

For example, you know that the mean for IQ scores is 100 and the standard deviation for IQ scores is 15 (because we told you this in the book and because you can see it by examining Figure 15.8).

Therefore, if your IQ is 115, you can get your z-score...

                       115   -   100               15
Z-score   =   --------------- =    --------   =    1
                              15                       15

An IQ of 115 falls one standard deviation above the mean.

Note that once you have a set of z-scores, you can convert to any other scale by using this formula: New score = Z-score(SD of new scale) + mean of the new scale.

For example, let’s convert a z-score of three to an IQ score
New score=3(15) + 100 (remember, the mean of IQ scores is 100 and the standard deviation of IQ scores is 15). Therefore, the new score (i.e., the IQ score converted from the z-score of 3 using the formula I just provided) is equal to 145 (3 times 15 is 45, and when 100 is added you get 145).

Examining Relationships Among Variables

We have been talking about relationships among variables throughout your textbook. For example, we have already talked about correlation (e.g., see Figure 2.2 on page 44), partial correlation (e.g., see page 341), analysis of variance which is used for factorial designs (e.g., see pages 286-291), and analysis of covariance (e.g., see pages 274-275 and pages 341-342).

At this point in this chapter on descriptive statistics, I introduce two additional techniques that you also can use for examining relationships among variables: contingency tables and regression analysis.

Contingency Tables

When all of your variables are categorical, you can use contingency tables to see if your variables are related.

A contingency table is a table displaying information in cells formed by the intersection of two or more categorical variables.
An example is shown in Table 15.6.

When interpreting a contingency table, remember to use the following two rules:

Rule One. If the percentages are calculated down the columns, compare across the rows.
Rule Two. If the percentages are calculated across the rows, compare down the columns.
When you follow these rule you will be comparing the appropriate rates (a rate is the percentage of people in a group who have a specific characteristic).
When you listen to the local and national news, you will often hear the announcers compare rates.
The failure of some researchers to follow the two rules just provided has resulted in misleading statements about how categorical variables are related; so be careful.

Regression Analysis

Regression analysis is a set of statistical procedures used to explain or predict the values of a quantitative dependent variable based on the values of one or more independent variables.

In simple regression, there is one quantitative dependent variable and one independent variable.
In multiple regression, there is one quantitative dependent variable and two or more independent variables.

On pages 455-459, I show you the components of the regression equations (e.g., the Y-intercept and the regression coefficients). Here are the important definitions:

Regression equation-The equation that defines the regression line (see Figure 15.9 in book and below).

Here is the simple regression equation showing the relationship between starting salary (Y or your dependent variable) and GPA (X or your independent variable) (two of the variables in the data set included with this chapter on page 435).

Ŷ = 9,234.56 + 7,638.85 (X)

The 9,234.56 is the Y intercept (look at the above regression line; it crosses the Y axis a little below $10,000; specifically, it crosses the Y axis at $9,234.56).
The 7,638.85 is the simple regression coefficient, which tells you the average amount of increase in starting salary that occurs when GPA increases by one unit. (It is also the slope or the rise over the run).
Now, you can plug in a value for X (i.e., starting salary) and easily get the predicted starting salary.
If you put in a 3.00 for GPA in the above equation and solve it, you will see that the predicted starting salary is $32,151.11
Now plug in another number within the range of the data (how about a 3.5) and see what the predicted starting salary is. (Check on your work: it is $35,970.54)

On pages 458-459, I show a multiple regression equation with two independent variables.

The main difference is that in multiple regression, the regression coefficient is now called a partial regression coefficient, and this coefficient provides the predicted change in the dependent variable given a one unit change in the independent variable, controlling for the other independent variables in the equation. In other words, you can use multiple regression to control for other variables (i.e., for what we called in earlier chapters statistical control).