How can two groups have the same average but very different variation?
Quick answer
The average describes a group’s center, not how close its values are to one another. For example, both 4, 5, 5, 6 and 1, 3, 7, 9 have a mean of 5, but their ranges are 2 and 8. Compare the individual values or a measure of spread, such as range or standard deviation, to see how much the groups vary. [S1]
The question, the details, the next step.
How can two groups have the same average but still be different? I’m comparing a few sets of numbers and thought the group with the higher values would have the higher average.
The average, or mean, is the sum of the values divided by how many values there are. It describes a group’s center, but it does not show how tightly the values cluster around that center. [S1]
Here is a paired example:
- Group A: 4, 5, 5, 6
- Group B: 1, 3, 7, 9
Each group has four values and a total of 20, so each has a mean of 20 ÷ 4 = 5. Yet Group A’s values stay close to 5, while Group B’s values are more spread out. The mean is the same; the pattern of individual values is not.
What’s a simple way to show how much more spread out the second group is?
Start with the range: subtract the smallest value from the largest. Range is defined as the difference between the largest and smallest data values. [S1]
For Group A, the range is 6 − 4 = 2. For Group B, it is 9 − 1 = 8. So Group B has a larger range, even though both means are 5.
You can also inspect how far each number sits from the mean. In Group A, the values are at most 1 away from 5. In Group B, the values are 2 or 4 away. That makes the contrast visible without changing the data. The range is a quick summary, though it uses only the two extreme values; it does not show the full shape of a group.
Would standard deviation tell me the same thing, and when should I use it?
Standard deviation is another measure of spread. For these groups, it would show greater variation in Group B, whose values lie farther from the shared mean of 5. [S1]
For a quick comparison, the range is easy to calculate and explain. If you need a different summary of spread, standard deviation may be useful. Make sure you use the same measure when comparing groups, and give the actual values when the reader needs to see the distribution. A mean and one spread measure summarize data; they do not reproduce every detail of it.
How could I explain this clearly in a short answer for class?
You could write: “Both groups have a mean of 5: Group A (4, 5, 5, 6) and Group B (1, 3, 7, 9). Their ranges are 2 and 8, respectively, so Group B is more spread out. The mean alone does not show that difference.”
To check your own comparison, first calculate each mean from its values. Then compare a spread measure, such as range, and look back at the individual values to see what the summary leaves out. In this example, the equal means and unequal ranges establish the point.
A practical checklist
Write down all the values in each group.
Calculate each mean by dividing the sum by the number of values.
Find each range by subtracting the smallest value from the largest.
Compare the individual values to see what the mean and range may hide.
Name the measure of spread you used when explaining the comparison.
A few more useful answers
Does a larger range always mean every value is farther from the mean?
No. Range compares only the largest and smallest values. Check the full set or another spread measure to understand how the values are distributed.
Can two groups have the same mean and the same range but still differ?
Yes. The values between the minimum and maximum can be arranged differently. Looking at the individual values or another summary can reveal differences that mean and range miss.
What should I include when comparing two groups?
Give the values or a useful summary, report the mean, and name a measure of spread. That makes clear both where the groups are centered and how much their values vary.
Sources
References used for this guide.
- [S1] Chapter 8 Key Concepts — Contemporary Mathematics openstax.org · Checked