Mean, Median, Mode, and Range Practice: A Complete Guide to Mastering Basic Statistics
Understanding the four fundamental measures of central tendency and spread—mean, median, mode, and range—is essential for anyone studying mathematics, science, economics, or even everyday decision‑making. Still, these concepts form the backbone of data analysis, allowing you to summarize large sets of numbers with just a few representative values. In this guide, we’ll walk through each measure, show you how to calculate them, and provide plenty of mean mode median and range practice problems to reinforce your skills. By the end, you’ll feel confident tackling any data set that comes your way.
What Are Mean, Median, Mode, and Range?
Before diving into calculations, it helps to know what each term represents:
- Mean (often called the average) is the sum of all values divided by the number of values.
- Median is the middle number when the data are ordered from smallest to largest. If there is an even number of observations, the median is the average of the two central numbers.
- Mode is the value that appears most frequently. A data set can have one mode, more than one mode, or no mode at all.
- Range measures the spread of the data by subtracting the smallest value from the largest value.
These four statistics are collectively known as measures of central tendency (mean, median, mode) and a measure of dispersion (range). Practicing them together helps you see how they complement each other when describing a data set.
How to Calculate Each Measure
Finding the Mean
- Add all numbers in the data set.
- Divide the total by the count of numbers.
Example: For the set 4, 8, 6, 5, 3, the sum is 26. There are 5 numbers, so the mean is 26 ÷ 5 = 5.2.
Determining the Median
- Arrange the numbers in ascending order.
- If the count is odd, the median is the middle number.
- If the count is even, take the average of the two middle numbers.
Example: For 7, 2, 9, 4, order them → 2, 4, 7, 9. With four numbers (even), the median is (4 + 7) ÷ 2 = 5.5.
Identifying the Mode
- Look for the value that occurs most often.
- If multiple values share the highest frequency, each is a mode (bimodal, trimodal, etc.).
- If all values appear only once, there is no mode.
Example: In 1, 2, 2, 3, 4, 4, 4, the number 4 appears three times, more than any other, so the mode is 4 The details matter here..
Computing the Range
- Identify the smallest (minimum) and largest (maximum) values.
- Subtract the minimum from the maximum.
Example: For 10, 3, 7, 12, the minimum is 3 and the maximum is 12, so the range is 12 – 3 = 9 Not complicated — just consistent..
Practice Problems: Mean, Median, Mode, and Range
Below are ten data sets. For each, calculate the mean, median, mode, and range. After you’ve attempted them, check the step‑by‑step solutions in the next section.
| # | Data Set |
|---|---|
| 1 | 5, 9, 12, 7, 3 |
| 2 | 14, 14, 8, 20, 6 |
| 3 | 1, 2, 2, 3, 4, 5, 5, 5 |
| 4 | 11, 13, 13, 11, 9 |
| 5 | 22, 18, 19, 21, 20, 22 |
| 6 | 4, 4, 4, 4, 4 |
| 7 | 7, 9, 10, 12, 13, 15 |
| 8 | 30, 25, 28, 27, 26, 31 |
| 9 | 5, 8, 8, 9, 10, 10, 12 |
| 10 | 100, 50, 75, 80, 60, 90 |
Step‑by‑Step Solutions
Problem 1: 5, 9, 12, 7, 3
- Mean: (5+9+12+7+3) = 36 → 36 ÷ 5 = 7.2
- Median: Ordered → 3, 5, 7, 9, 12 → middle = 7
- Mode: No repeats → no mode
- Range: Max 12 – Min 3 = 9
Problem 2: 14, 14, 8, 20, 6
- Mean: (14+14+8+20+6) = 62 → 62 ÷ 5 = 12.4
- Median: Ordered → 6, 8, 14, 14, 20 → middle = 14
- Mode: 14 appears twice → 14
- Range: 20 – 6 = 14
Problem 3: 1, 2, 2, 3, 4, 5, 5, 5
- Mean: Sum = 27 → 27 ÷ 8 = 3.375
- Median: Ordered list already; 8 numbers → average of 4th (3) and 5th (4) → (3+4)/2 = 3.5
- Mode: 5 appears three times → 5
- Range: 5 – 1 = 4
Problem 4: 11, 13, 13, 11, 9
- Mean: (11+13+13+11+9) = 57 → 57 ÷ 5 = 11.4
- Median: Ordered → 9, 11, 11, 13, 13 → middle = **1
Problem 4 (continued): 11, 13, 13, 11, 9
- Mean: (11 + 13 + 13 + 11 + 9) = 57 → 57 ÷ 5 = 11.4
- Median: Arranged as 9, 11, 11, 13, 13, the central value is 11
- Mode: The number 11 and 13 each occur twice, so both are modes (bimodal).
- Range: Largest – smallest = 13 – 9 = 4
Problem 5: 22, 18, 19, 21, 20, 22
- Mean: (22 + 18 + 19 + 21 + 20 + 22) = 132 → 132 ÷ 6 = 22.0 (or simply 22)
- Median: Sorted order → 18, 19, 20, 21, 22, 22. With six items, the median is the average of the third and fourth numbers: (20 + 21) ÷ 2 = 20.5
- Mode: 22 appears twice, while every other value appears once, so the mode is 22
- Range: Maximum = 22, Minimum = 18 → 22 – 18 = 4
Problem 6: 4, 4, 4, 4, 4
- Mean: (4 × 5) ÷ 5 = 4
- Median: The ordered list is all 4’s; the third term (the middle of five) is 4
- Mode: Every occurrence is identical, giving the single mode 4
- Range: Same minimum and maximum (both 4) → 4 – 4 = 0 (a zero range indicates no spread)
Problem 7: 7, 9, 10, 12, 13, 15
- Mean: (7 + 9 + 10 + 12 + 13 + 15) = 70 → 70 ÷ 6 ≈ 11.67 (rounded to two decimals)
- Median: Sorted list already; with six numbers the median is (10 + 12) ÷ 2 = 11
- Mode: All values are unique, so there is no mode
- Range: Highest = 15, lowest = 7 → 15 – 7 = 8
Problem 8: 30, 25, 28, 27, 26, 31
- Mean: (30 + 25 + 28 + 27 + 26 + 31) = 176 → 176 ÷ 6 ≈ 29.33
- Median: Ordering gives 25, 26, 27, 28, 30, 31; the median is again (27 + 28) ÷ 2 = 27.5
- Mode: Each number appears only once, therefore no mode exists.
- Range: Largest = 31, smallest = 25 → 31 – 25 = 6
Problem 9: 5, 8, 8, 9, 10, 10, 12
- Mean: (5 + 8 + 8 + 9 + 10 + 10 + 12) = 71 → 71 ÷ 7 ≈ 10.14
- Median: Sorted sequence → 5, 8, 8, 9, 10, 10, 12; the fourth (middle) value is 9
- Mode: The numbers 8 and 10 each appear twice, which is the highest frequency, so the dataset is bimodal with modes 8 and 10
- Range: Maximum = 12, minimum = 5 → 12 – 5 = 7
Problem 10: 100, 50, 75, 80, 60, 90
- Mean: (100 + 50 + 75 + 80 +
