Mean, median, interquartile range, and standard deviation are four of the most important tools for describing a data set. The mean and median describe the center. The IQR and standard deviation describe spread. The key is knowing what each measure tells you and when one is more useful than another.
This practice guide gives worked examples, a quick comparison table, 16 original problems, and a full answer key.
Quick Reference
| Measure | What It Describes | Best Use |
|---|---|---|
| Mean | Average value | Data without strong outliers |
| Median | Middle value | Skewed data or data with outliers |
| IQR | Spread of the middle 50% | Comparing variability when outliers exist |
| Standard deviation | Typical distance from the mean | Roughly symmetric data or normal distribution contexts |
Worked Example 1: Mean and Median
Data set: 6, 9, 10, 12, 18
Mean: Add all values and divide by the number of values.
(6 + 9 + 10 + 12 + 18) / 5 = 55 / 5 = 11
Median: The data are already ordered. The middle value is 10.
The mean is higher than the median because 18 pulls the average upward.
Worked Example 2: IQR
Data set: 3, 5, 7, 8, 10, 12, 15
The median is 8. The lower half is 3, 5, 7, so Q1 = 5. The upper half is 10, 12, 15, so Q3 = 12.
IQR = Q3 – Q1 = 12 – 5 = 7.
Worked Example 3: Standard Deviation Idea
Compare these two data sets:
- Set A: 8, 9, 10, 11, 12
- Set B: 2, 6, 10, 14, 18
Both have mean 10. But Set B is much more spread out. That means Set B has the larger standard deviation. You do not always need to compute the exact standard deviation to understand the concept: bigger typical distance from the mean means bigger standard deviation.
Worked Example 4: Calculating Standard Deviation
For a short introductory example, use the data set 4, 6, 8. The mean is 6. The deviations from the mean are -2, 0, and 2. Square the deviations to get 4, 0, and 4. The average squared deviation is 8 / 3, so the population standard deviation is sqrt(8 / 3), about 1.63.
The main idea is more important than the arithmetic: standard deviation is built from distances from the mean. Values farther from the mean create larger squared deviations and a larger standard deviation.
Practice Problems
1. Find the mean of 4, 6, 8, 12, 15.
2. Find the median of 13, 7, 9, 21, 10, 12.
3. Find the range of 5, 8, 8, 11, 19, 20.
4. Find Q1, Q3, and IQR for 2, 4, 5, 7, 9, 11, 14.
5. Find Q1, Q3, and IQR for 10, 12, 15, 18, 20, 25, 30, 34.
6. A data set has mean 50. A new value of 100 is added. Will the mean increase, decrease, or stay the same?
7. Which is more affected by an outlier: mean or median?
8. Which data set has the larger standard deviation: A = 9, 10, 11 or B = 2, 10, 18?
9. A box plot has minimum 4, Q1 9, median 12, Q3 20, and maximum 28. What is the IQR?
10. The test scores 70, 72, 75, 80, 98 have mean 79. Which score is most responsible for pulling the mean above the median?
11. A data set has standard deviation 0. What must be true about all values in the data set?
12. In a skewed income data set with a few very high salaries, which measure of center is usually more representative: mean or median?
13. The data set 3, 3, 4, 5, 20 has mean 7. Which value is the outlier?
14. Two classes have the same mean quiz score. Class A scores are tightly clustered from 78 to 84. Class B scores range from 55 to 100. Which class has the larger standard deviation?
15. Find the median and IQR of 6, 8, 8, 10, 12, 15, 18.
16. A teacher wants to describe the typical score on a test where most scores are 70-85 but one score is 20. Should the teacher use the mean or median? Explain.
Answer Key
1. Sum = 4 + 6 + 8 + 12 + 15 = 45. Mean = 45 / 5 = 9.
2. Order the data: 7, 9, 10, 12, 13, 21. Median = (10 + 12) / 2 = 11.
3. Range = 20 – 5 = 15.
4. Median is 7. Lower half: 2, 4, 5 -> Q1 = 4. Upper half: 9, 11, 14 -> Q3 = 11. IQR = 11 – 4 = 7.
5. Lower half: 10, 12, 15, 18 -> Q1 = (12 + 15) / 2 = 13.5. Upper half: 20, 25, 30, 34 -> Q3 = (25 + 30) / 2 = 27.5. IQR = 27.5 – 13.5 = 14.
6. The mean will increase because 100 is above the current mean of 50.
7. The mean is more affected by an outlier because every value is included in the average.
8. Set B has the larger standard deviation because its values are farther from the mean.
9. IQR = Q3 – Q1 = 20 – 9 = 11.
10. The score 98 pulls the mean upward.
11. If standard deviation is 0, all values are identical.
12. The median is usually more representative because it resists extreme high values.
13. The outlier is 20. It is much farther from the rest of the data and pulls the mean above most values.
14. Class B has the larger standard deviation because its scores are more spread out from the mean.
15. The median is 10. Lower half: 6, 8, 8, so Q1 = 8. Upper half: 12, 15, 18, so Q3 = 15. IQR = 15 – 8 = 7.
16. The median is better because the score of 20 is a low outlier that pulls the mean downward.
Common Mistakes
Using the Mean When the Median Is Better
If a data set has a strong outlier, the mean can be misleading. For example, in home prices or salaries, a few very high values can pull the mean upward. The median often gives a clearer picture of a typical value.
Confusing Range and IQR
Range uses the maximum and minimum. IQR uses Q3 and Q1. If the question asks about the spread of the middle 50%, it is asking for IQR, not range.
Thinking Standard Deviation Is Always a Formula Problem
Many middle school, high school, placement, and test-prep questions ask about standard deviation conceptually. If one data set is clustered tightly and another is spread out, the spread-out data set has the larger standard deviation.
Diagnostic Checklist
Before moving on, check whether you can answer these three questions for any data set: What measure describes center? What measure describes spread? Are there outliers or skew that should change the measure you choose? If the answer involves a “typical” value with an outlier, think median. If it involves the middle 50% or box plots, think IQR. If it involves typical distance from the mean, think standard deviation.
Short Study Plan
Day 1 should focus on ordering data and finding mean and median. Day 2 should focus on quartiles, box plots, and IQR. Day 3 should compare data sets by spread. Day 4 should mix all four measures and require a sentence explaining which measure fits the context. That final explanation is where many statistics questions become easier.
Next Study Step
After you can compute these measures, practice choosing the right measure for a context. Ask: Is the data skewed? Are there outliers? Am I describing center or spread? That decision is often the real statistics skill.
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