Descriptive statistics describe the typical and basic features of a data set. They also form the basis of more complex statistics we will examine later.
We can divide descriptive statistics into two categories: The typical score, also called “central tendency,” and the differences in scores, or the “variability.”
Let's look closer at the measurement of central tendency. This is comprised of the mean, or average score; the median, which is the middle value or 50th percentile; and the mode, the most common score.
Now let's look at how the mean, median, and mode are each calculated. It is rou useful to be able to quickly do this for a small range of numbers, but for larger data sets we use a computer program for the calculations.
Let's start with the mean. Consider the following five scores: 1, 2, 4, 5, 8. To calculate the average score we add up all the numbers: 1 plus 2 plus 4 plus 5 plus 8 equals 20. Now divide your total by how many scores there were. In this case, there were 5 scores. 20 divided by 5 equals 4. The mean is 4.
[ An activity to practice calculating the mean.]
1 + 3 + 5 + 7 + 9 + 11 = [ ___ ]
___ ÷ ___ = ___
Resolution to practice activity for calculating the mean.
1 + 3 + 5 + 7 + 9 + 11 = 36
36 ÷ 6 = 6
In APA style we can use the shorthand upper case M in italics to represent mean.
1 + 2 + 4 + 5 + 8 = 20
20 ÷ 5 = 4
mean notation
M = 4
It's easy to calculate the median with an odd number of scores, like you see here
[next screen visual].
4, 4, 5, 7, 10
It is simply the middle value, so here it would be five.
4, 4, 5, 7, 10, 12
But for an even number of scores, simply take the average of the two middle- most scores. Here this would be five and seven.
4, 4, 5, 7, 10, 12
(5 + 7) ÷ 2 = 6
So, we add 5 and 7, divide it by two to get the average, and come up with our median, 6.
[An activity to practice calculating the median.]
1, 3, 4, 6, 9, 11
( ___ + ___ ) ÷ 2 = ___
Resolution to practice activity for calculating the median.
1, 3, 4, 6, 9, 11
(4 + 6) ÷ 2 = 5
In APA style we can use the shorthand uppercase M followed by lowercase dn in italics to represent median.
median notation
Mdn = 5
Finally, let's look at the mode, or the most common score.
4, 4, 5, 7, 10
With this set of scores, obviously the most common is four, as it appears twice.
1, 6, 8, 14, 20
Sometimes, there is no mode, like you see in this set of scores (above).
[An activity to practice calculating the mode.]
1, 3, 4, 6, 9, 9, 11
[Resolution to practice activity for calculating the mode.]
1, 3, 4, 6, 9, 9, 11
In APA style there is no agreed upon shorthand for modes.
Now that we've looked at central tendency, let's focus on variability, or how to quantify the differences in scores. Reporting how much scores vary is as important as reporting central tendency.
The mode, median, and mean do not describe variation. Instead, standard deviation and range are two ways of reporting variation.
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
Consider three groups (above).
Do these groups differ?
Yes or No
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
What is the mean for each group?
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Answer: ___
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Answer: ___
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
Answer: ___
What is the median for each group?
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Answer: ___
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Answer: ___
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
Answer: ___
What is the mode for each group?
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Answer: ___
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Answer: ___
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
Answer: ___
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
mean, median, mode = 8
The groups differ but the mean, median, and mode cannot show this. We need to quantify the variation.
We can use range to report this variation. Range is the difference between the highest and lowest score. 100% of the scores fall within the number of scales units indicated by the range.
range
4, 6, 8, 12, 15, 16, 18, 21, 25, 34, 37
37- 4 = 33
Consider this group of scores (above). The range of these scores is the highest score minus the lowest score. Or, 37 minus four, giving us the range of 33.
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Range = 0
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
So, to return to our groups, if we look at Group 1, 100% of these scores fall at the same scale value. The range is therefore zero.
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Range = 0
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Range = 4
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
In Group 2, we take the highest score of 10 and subtract the lowest score of six from it to get our range of four. 100% of the scores fall within four scale units of each other.
What is the range for Group 3?
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
Answer: ____
[Correct answer is 14.]
A problem with range is that it does not utilize all the scores; only the smallest and largest score are used to calculate the range.
Group 1: 2, 20, 20, 21, 21, 22, 22, 200
Group 2: 2, 31, 62, 86, 119, 148, 185, 200
For example, these two groups both have a range of 198 (200 – 2), but Group 1 has much variation in scores than Group 2. In Group 1 the mean, median, and mode do a better job of predicting the typical score than in Group 2. However, just knowing the range does not inform how good a predictor the mean, median, and mode are.
For this reason, although the range provides useful information, the standard deviation is often preferred as a measure of variation.
Although it is possible to calculate standard deviation by hand, it is time consuming and so not discussed here.
The key thing to understand is, the more variation there is in scores, the larger the standard deviation.
Group 1: 8, 8, 8, 8, 8, 8, 8, 8
Group 2: 9, 8, 6, 10, 7, 8, 8, 8
Group 3: 15, 1, 14, 2, 8, 8, 12, 4
In our sample distributions, Group 1 has a standard deviation of 0. Group 2 has a standard deviation of 1.20. Group 3 has a standard deviation of 5.37.
As we will see as we progress through this course, although statistics are factual (in that they report what the data/numbers show), it is possible to manipulate statistics to work in your favor. One way of doing this is by which statistics you choose to report.
To show this, let's look at the use of the descriptive statistics (mean and median) in the context of the 1994 baseball strike.During this strike about salary, the argument of the owners was that the typical salary was $1.2 million. The players, on the other hand, claimed the average salary was $415,000.
Sample Salary (,000) for 1994 baseball strike:
225, 275, 390, 390, 395, 415, 510, 760, 2.6, 5.8
players: $415,000
owners: $1,200,000
If you look at a sample of salaries, what measure of tendency was each side using?
What measure is each group using?
Choose mean, median or mode.
Sample Salary (,000) for 1994 baseball strike:
225, 275, 390, 390, 395, 415, 510, 760, 2.6, 5.8
players: $415,000
Answer: _________
owners: $1,200,000
Answer: _________
Answers:
players: median
owners: mean