Mathematics › Statistics
Master the four fundamental measures that let you summarize, compare, and interpret any set of numbers — with step-by-step worked examples for every concept.
Middle School High School Data AnalysisEvery day, people summarize data to make sense of the world. A teacher wants to know how well her class did on a test. A basketball coach compares players' scoring averages. A scientist measures the same chemical reaction multiple times and wants one representative number. These are all problems that the four measures of basic statistics — mean, median, mode, and range — were designed to solve.
Together, mean, median, and mode are called measures of central tendency because they each describe the "center" or "typical value" of a data set. Range is a measure of spread — it tells you how far apart the values are. Knowing when to use each one is just as important as knowing how to calculate it.
Sum of all values divided by how many there are
Middle number when data is sorted in order
The value that appears most often
Largest value minus smallest value
The mean is what most people think of when they hear the word "average." To find it, add up every number in your data set, then divide by how many numbers there are.
The mean works best when your data has no extreme outliers — numbers that are much higher or lower than the rest. If there are outliers, they drag the mean away from where most data points cluster, making it less representative.
A student scored the following on six math quizzes: 78, 85, 90, 88, 76, 93
Step 1: Add all the scores.
78 + 85 + 90 + 88 + 76 + 93 = 510
Step 2: Divide by the number of scores (6).
510 ÷ 6 = 85
Mean = 85 — the student averages 85 points per quiz.
Seven employees earn these annual salaries (in thousands): $32, $35, $38, $40, $42, $45, $200
Step 1: Sum = 32 + 35 + 38 + 40 + 42 + 45 + 200 = 432
Step 2: Mean = 432 ÷ 7 = $61,714
But six of the seven employees earn less than $45,000. The $200,000 salary (the owner) pulled the mean up to $61,714 — not a fair representation of what a typical employee earns. This is why we also need the median.
Sometimes different values count more than others. In a class where tests are worth 60% of the grade and homework is worth 40%, you use a weighted mean: multiply each value by its weight, add the products, and divide by the total weight. This concept appears in GPA calculations, final exam grading, and many real-world situations.
The median is the middle value of a sorted data set. It is resistant to outliers — extreme values do not change it — which makes it a better measure of center when data is skewed.
Find the median of: 14, 7, 21, 3, 18, 9, 25
Step 1: Sort from smallest to largest.
3, 7, 9, 14, 18, 21, 25
Step 2: Count the values. There are 7 (odd). The middle position is (7+1)/2 = 4th.
3, 7, 9, 14, 18, 21, 25
Median = 14
Find the median of: 12, 5, 19, 8, 23, 16
Step 1: Sort from smallest to largest.
5, 8, 12, 16, 19, 23
Step 2: Count the values. There are 6 (even). The two middle positions are 3rd and 4th.
5, 8, 12, 16, 19, 23
Step 3: Average the two middle values.
(12 + 16) ÷ 2 = 28 ÷ 2 = 14
Median = 14
Notice that in Example 2 (salaries), the median would be the 4th value after sorting: $32, $35, $38, $40, $42, $45, $200 — so median = $40,000. That is far more representative of a typical employee's pay than the mean of $61,714.
The mode is the value that appears most frequently in a data set. Unlike mean and median, the mode can be used with non-numerical (categorical) data — for example, the most common eye color in a classroom or the most popular pizza topping.
Data set: 4, 7, 2, 7, 9, 3, 7, 5, 2, 7
Count each value:
Mode = 7
Shoe sizes in a class: 6, 7, 8, 7, 9, 8, 10, 6, 7, 8, 11, 7, 8
Size 7 appears 4 times. Size 8 appears 4 times. All others appear fewer times.
This data set is bimodal: Mode = 7 and 8
The range measures how spread out a data set is. It is simply the difference between the largest and smallest values. A small range means data is tightly clustered; a large range means data is spread widely.
Daily high temperatures for a week (°F): 72, 68, 75, 80, 65, 78, 71
Maximum = 80°F, Minimum = 65°F
Range = 80 − 65 = 15°F
Interpretation: Over the week, temperatures varied by 15 degrees.
The range is easy to calculate but sensitive to outliers — one extremely high or low value dramatically changes it. More advanced statistics use measures like interquartile range (IQR) or standard deviation to describe spread more robustly, but range is the perfect starting point.
Real statistical analysis uses all four measures together to paint a full picture of the data.
A basketball player scored the following points in 9 games: 18, 22, 15, 31, 19, 22, 14, 18, 22
Step 1: Find the Mean
Sum = 18+22+15+31+19+22+14+18+22 = 181
Mean = 181 ÷ 9 = 20.1 points per game
Step 2: Find the Median
Sorted: 14, 15, 18, 18, 19, 22, 22, 22, 31
9 values → 5th value is the middle: Median = 19
Step 3: Find the Mode
22 appears 3 times (most frequent): Mode = 22
Step 4: Find the Range
Range = 31 − 14 = 17 points
Interpretation: The player scores around 19–22 points most games (median and mode agree). The mean of 20.1 is close to these, suggesting the data is fairly symmetric. The 31-point game was a high outlier but did not distort the picture much. The range of 17 shows moderate game-to-game variation.
| Situation | Best Measure | Why |
|---|---|---|
| Symmetric data, no outliers | Mean | Uses every value, most precise summary |
| Skewed data or outliers present | Median | Resistant to extreme values |
| Categorical data (colors, brands) | Mode | Only measure that works on non-numbers |
| Most popular or common item | Mode | Directly identifies the most frequent value |
| Describing variation or spread | Range | Captures full spread from min to max |
| Home prices, incomes | Median | A few very expensive homes or very high earners distort the mean |
Problem 1: A student's test scores are 84, 91, 78, 95, and 87. Find the mean, median, mode, and range.
Problem 2: Eight runners finish a race in these times (seconds): 48, 52, 45, 61, 48, 55, 49, 48. Find all four measures.
Problem 3: A data set has a mean of 20 and contains the values 14, 22, 18, 25, and one unknown value. Find the missing value.
Problem 4: The salaries at a company are $28K, $32K, $35K, $36K, $38K, $40K, and $150K. Which measure of center best represents a typical employee salary?
Problem 5: Determine whether this data set is unimodal, bimodal, or has no mode: 3, 5, 7, 5, 9, 3, 11, 7, 5, 3