### Descriptive Statistics

What are descriptive statistics? They are just want they sound like -- statistics that describe a group of numbers. Basically, they take a lot of numbers and provide a meaningful summary with a single number. Consider the 26 numbers below.

2, 3, 4, 5, 6, 5, 4, 2, 4, 5, 6, 6, 7, 4, 5, 5, 7, 8, 9, 8, 7, 6, 5, 4, 2, 1,

Everyone is familiar with the average or mean. Well the mean for these 26 numbers is 5 and it is a descriptive statistic. We have just taken a bunch of numbers and described them with an easy to understand statistic. In this section we will cover the basics of how such statistics should be reported in APA style.

First, when reporting statistics we simply state them. It is not a time to be eloquent or creative. Simplicity always wins out in the end. Be concise. Do not make value judgments. Thus, you don’t report statistics as “good,” “bad,” “uninteresting,” etc. The statistical results are the statistical results. You just report them. Later you can interpret them and draw conclusions.

Next, when you report statistics you can do it completely using words or you can include statistical symbols and abbreviations. For example, you might say

The sample consisted of 100 college age students and was 49% male and 51% female.

Or you might say it with using one or more statistical symbols.

The sample of college students was 49% male and 51% female (*N* = 100).

In the example above N is the symbol used in APA style to denote the total sample size. There are a lot of symbols that you might use to represent basic descriptive statistics when writing up results. The most commonly used for descriptive statistics are shown below.

Symbol/Abbreviation |
Description |

n |
Number of cases in a subsample |

N |
Total number of Cases |

M |
Sample mean |

Mdn |
Sample median |

SD |
Standard Deviation |

Note that all of these should appear in italic font type.

If you are going to use symbols and abbreviations you need to be mindful of the spacing. Just as this is difficulttoread so is the presentation of mathematical equations and results when there are no spaces. Thus, don’t write this

Wrong: *M*=5.10,*SD*=1.72.

Instead it should be this with spaces

*M* = 5.10, *SD* = 1.72.

Once again as the writer you need to make it as easy as possible on your reader.

Another practice to avoid is writing narratives using statistical symbols within the textual description. Thus, do not write this.

Wrong: The *M*s for extroversion and agreeableness were 3.10 and 2.60 respectively.

Instead it is easier on the reader and the accepted standard to write it like this.

The means for extroversion and agreeableness were 3.10 and 2.60 respectively.

There is one file practice to note before we move on to an example. Only use the % symbol when it is immediately preceded by a numeral. How I just used it was wrong. I should have spelled out “percentage.”

We first calculated the percent correct for group 1 and group 2. The results showed it was 89% for group 1 and 79% for group 2.

### Descriptive Statistics Example

Generally when doing statistical analyses one of the first things you will do is describe your sample and the results for the measures you are using so let’s work through a sample scenario.

The Wonderlic Cognitive Ability Test is a commonly used test for employee selection. It was first created in 1936 by E. F. Wonderlic, but has been updated over the years. The test is simple consisting of 50 questions that must be completed in 12 minutes and this may be part of the reason it has been used so much being given to millions of employees and potential hires. The results show that the Wonderlic is a worthwhile measure of cognitive ability that can be successfully used for hiring employees in a variety of occupations.

For practice we are going to suppose that you had access to employees in three different occupations and that you administered the Wonderlic to each group. You entered the data into your favorite statistical program and then obtain the table shown below. How should you write up the basic description of this information?

The Wonderlic Cognitive Ability Test was given to a sample of 54 individuals who were currently employed as an accountant (*n* = 22), an electrical engineer (*n* = 19), or an investment analyst (*n* = 13). The descriptive statistics show that the mean score for electrical engineers was 28.53 (*SD* = 3.19), for accountants it was 27.32 (*SD* = 3.93), and for investment analysts it was 25.92 (*SD* = 5.66).

Of course this is just one way to say all of this. There are countless other ways.

Fifty-four people who were employed in one of three occupations were assessed using the Wonderlic Cognitive Ability Test. Of these examinees 41% were accountants (*n* = 22), 35% where electrical engineers (*n* = 19), and 24% where investment analysts (*n* = 13). The descriptive statistics showed that electrical engineers tended to score the highest (*M* = 28.53, *SD* = 3.19), followed by accountants (*M* = 27.32, *SD* = 3.93), and then investment analysts (*M* = 25.92, *SD* = 5.66).

A test of cognitive ability (Wonderlic) was given to a sample of 22 accounts, 19 electrical engineers, and 13 investment analysts (*N* = 54). Electrical engineers scored the highest with a mean of 28.53 and a standard deviation of 3.19. Accountants scored between the other two occupations with a mean score of 27.32 and a standard deviation of 3.93. Investment analysts had the lowest mean score of 25.92 with a standard deviation of 5.66.

### z Scores

The z score can be a descriptive or inferential statistic. In textbooks they are often first presented as a descriptive statistic and then the books move on to the inferential use of z scores (although they don’t always make this distinction clear). In real life, at least in my real life experiences, I have used them exclusively as descriptive statistics to describe where a particular test score lies within a normal distribution. Therefore, I have included them in this descriptive statistic section.

What is the correct symbol to use? It is a lowercase z that is in italic font. The symbol is all that is really new here as you follow all the other rules already presented for the other descriptive statistics.

There is one other writing practice you should be aware of, however. Since there is no other way to spell out “z” as it is only a letter you might try to start a sentence with “z.” What do you do then? Do you capitalize it, sentences are should start with a capital letter, or do you keep the lowercase z as the first letter? Neither. You should reword the sentence to avoid starting with “z.”

Wrong: z scores for the first three participants were 0.98, 1.34, and -0.02.

Instead it would be best to reword the sentence.

The first three participants had z scores of 0.98, 1.34, and -0.02.

There are a few things to note in the example directly above.

First, you should use leading zeros if the z score is between -1.00 and 1.00 because z scores can have an absolute value greater there 1.00.

Second, do NOT place a hyphen between z and score. Probably half the time you will see it written with a hyphen. Different publishers and editors do not always perfectly follow APA style. They may have their own rules that they consistently apply. Technically this is not consistent with APA style. My reading of the manual indicates not to use a hyphen. And every occurrence in the APA publication manual and in their online blog about APA style fails to place a hyphen between z and score.

Third, be careful because word processes often like to auto-capitalize the z.

Here are a few more examples of reporting z scores.

The test taker in question scored in the top 10% of all test takers (z = 1.66).

After the tests were equated z scores were calculated and it was found that the subject had a z score of 0.23.

The obtained z score of -1.45 showed that Johnny scored below the expected proficiency for a fifth grader.

Johnny’s test results showed below average fifth grade math proficiency (z = -1.45).

### Test Yourself

** 1. Five of the respondents strongly disagreed with the statement.**

%

percent

**2. Which of the following is correct?**

The mean was 5.00 and standard deviation was 1.23.

The *M* was 5.00 and the *SD* was 1.23.

**3. Which is easier to read?**

*M*=27.22, *SD*=10.02

*M* = 27.22, *SD* = 10.02

**4. What does N represent?**

Subsample size

Total sample size

Either subsample or total sample size

**5. The mean score for Group 1 was the largest ( = 4.87)**

Mean

M

*M*