What is the difference between mean and median?

The mean is the arithmetic average of all values. The median is the middle value when data is sorted. With skewed data or outliers, the median is often a more representative measure.

When should I use mode instead of mean or median?

Mode is most useful for categorical data (e.g., most popular color) or when you want to know the most frequent value. It's the only measure of center that works for non-numeric data.

What does range tell us about data?

Range = maximum − minimum. It measures the total spread of the data but is sensitive to outliers. A single extreme value can make range misleading.

Statistics 101: Mean, Median, Mode & More

Why Statistics Matter

Every day, numbers compete for your attention: sports averages, poll results, test scores, temperature forecasts. Statistics gives you the tools to look past individual data points and see the shape of information. Before you can tackle complex topics like standard deviation or regression, you need to master three foundational measures: mean, median, and mode, plus the basic spread metric range. These concepts appear in middle school math, high school algebra, college entrance exams, and virtually every scientific paper ever published.

This lesson walks you through each measure from scratch, shows when to prefer one over another, and builds toward reading and interpreting simple data displays.

Step-by-Step Lesson

Mean (Arithmetic Average)

The mean is what most people mean when they say "average." Add up all the values, then divide by how many values there are.

Mean = (Sum of all values) ÷ (Number of values)

Worked Example — Test Scores

A student received these scores on five quizzes:

72 85 90 78 95

Step 1 — Sum: 72 + 85 + 90 + 78 + 95 = 420

Step 2 — Divide by count: 420 ÷ 5 = 84

The mean score is 84.

Key insight: Every value contributes equally to the mean. A single very high or very low value (an outlier) can pull the mean away from the "typical" value significantly.

Median (Middle Value)

The median is the middle value of an ordered data set. It is resistant to outliers, making it better than the mean when data is skewed.

Procedure:

Sort the data in ascending order.
If n is odd: the median is the value at position (n + 1) / 2.
If n is even: the median is the mean of the two middle values.

Worked Example — Odd Count (7 values)

37912141722

Position = (7 + 1) / 2 = 4th value = 12. Median = 12.

Worked Example — Even Count (6 values)

5811142025

Two middle values: 11 and 14. Median = (11 + 14) / 2 = 12.5.

Mode (Most Frequent Value)

The mode is the value that appears most often. A data set can have no mode, one mode (unimodal), two modes (bimodal), or more (multimodal).

Worked Example

479793972

Count each: 4→1, 7→3, 9→3, 3→1, 2→1.

Both 7 and 9 appear 3 times — this data set is bimodal: mode = 7 and 9.

Mode is especially useful for categorical data. "What size shoe is purchased most often?" has a mode, not a meaningful mean.

Range (Spread of Data)

Range is the simplest measure of spread (variability) in a data set.

Range = Maximum value − Minimum value

Worked Example

184329671152

Max = 67, Min = 11. Range = 67 − 11 = 56.

Range is quick but fragile. One extremely large or small outlier inflates the range without telling you much about the typical spread. A class of 25 students where 24 score 80–95 but one scores 12 has a misleadingly large range.

Choosing the Right Measure

This is where statistics becomes judgment, not just calculation. The right measure depends on the shape of your data and the question being asked.

Situation	Best Measure	Why
Symmetric, no outliers	Mean	Uses all data efficiently
Skewed data or outliers present	Median	Not distorted by extreme values
Categorical data	Mode	Only applicable measure
Typical vs. extreme comparison	Mean + Median together	Gap reveals skew direction
Spread / variability	Range (as a first estimate)	Quick, easy to communicate

Key insight: When mean > median, data is right-skewed (a few very high values pull the mean up). When mean < median, data is left-skewed. Income data is famously right-skewed: a few billionaires make the national mean income much higher than what most people actually earn.

Outliers and Their Impact

An outlier is a value that lies far from the rest of the data. Outliers can be legitimate (an unusually tall person) or errors (a data-entry mistake). Before summarizing data, always look for outliers and decide how to handle them.

Outlier Effect Example

Monthly salaries at a small company (in $1,000s):

4548505253310

With outlier: Mean = (45+48+50+52+53+310) / 6 = 558 / 6 = $93K. Median = (50+52)/2 = $51K.

Without outlier: Mean = (45+48+50+52+53) / 5 = 248 / 5 = $49.6K. Median = $50K.

The CEO's $310K salary inflates the mean by 88% but barely moves the median. Here, median is the fairer summary.

Practice Problems

Problem 1

Find the mean, median, and mode of: 8, 3, 5, 3, 7, 9, 3, 6.

Sort: 3, 3, 3, 5, 6, 7, 8, 9 (n = 8)

Mean: (3+3+3+5+6+7+8+9) / 8 = 44 / 8 = 5.5

Median: (5+6)/2 = 5.5 (4th and 5th values)

Mode: 3 appears 3 times — mode = 3

Note: mean equals median here, suggesting the data (excluding mode) is roughly symmetric.

Problem 2

A list of daily temperatures (°F) for a week: 68, 72, 71, 69, 88, 70, 72. Find the range and explain whether the mean or median better represents typical daily temperature.

Range: 88 − 68 = 20°F

Sort: 68, 69, 70, 71, 72, 72, 88

Mean: (68+72+71+69+88+70+72) / 7 = 510 / 7 ≈ 72.9°F

Median: 4th value = 71°F

Better measure: The 88°F day is an outlier (unusually hot). The median (71°F) better reflects the typical temperature; the mean is pulled up by the one hot day.

Problem 3

A shoe store sells these sizes in one hour: 7, 9, 8, 10, 9, 7, 9, 8, 11, 9. What is the mode, and why is it more useful than the mean for the store manager?

Tally: 7→2, 8→2, 9→4, 10→1, 11→1

Mode = 9 (appears 4 times)

Mean: (7+9+8+10+9+7+9+8+11+9) / 10 = 87 / 10 = 8.7

The mean of 8.7 does not correspond to any real shoe size. The mode (size 9) tells the manager which size to stock the most. Mode wins for categorical/discrete inventory decisions.

Problem 4

Data: 15, 22, 18, 25, 19, 23, 17. Calculate all four statistics (mean, median, mode, range).

Sort: 15, 17, 18, 19, 22, 23, 25 (n = 7)

Mean: (15+22+18+25+19+23+17) / 7 = 139 / 7 ≈ 19.86

Median: 4th value = 19

Mode: Each value appears once — no mode

Range: 25 − 15 = 10

Mean ≈ median indicates roughly symmetric data — no strong skew.

Problem 5

The mean of five numbers is 16. Four of the numbers are 12, 18, 14, and 20. What is the fifth number?

Mean = Sum ÷ Count → Sum = Mean × Count = 16 × 5 = 80

Sum of known values: 12 + 18 + 14 + 20 = 64

Fifth number = 80 − 64 = 16

This "reverse mean" technique is useful in competitions and tests.

5 Common Mistakes

1

Forgetting to sort before finding the median
The median requires an ordered list. Finding the "middle" of an unsorted list gives a random value, not the true median.
2

Assuming the mean is always the best average
In skewed distributions or data with outliers, the median is more representative. Always inspect the data before choosing.
3

Miscounting n (the number of values)
Adding one extra value or missing one changes both the mean and the median position calculation. Count carefully, especially when values repeat.
4

Declaring "no mode" when values tie
If two values both appear the most (and equally), the data is bimodal — both are modes. Only when every value appears exactly once is there truly no mode.
5

Confusing range with spread in general
Range only measures total span. Two data sets can have the same range but very different distributions. Range is a starting point, not a complete picture of variability.

Further Resources

Khan Academy: Data & Statistics

khanacademy.org

Mean, Median, Mode — Purple Math

purplemath.com

BBC Bitesize: Averages

bbc.co.uk

Mean — MathWorld Wolfram

mathworld.wolfram.com

Mean vs. Median — Statistics How To

statisticshowto.com