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Statistics guide

Mean, median, mode and standard deviation

How to find mean, median, mode and range, when each one is the right summary, and how to calculate and interpret standard deviation step by step.

Statistics starts with summarizing data: what is a typical value, and how spread out are the values? This guide covers the measures of center (mean, median, mode) and spread (range, interquartile range, standard deviation), plus how to choose the right one for a data set.

Measures of center

  • Mean: add all values and divide by how many there are. It uses every value, so extreme values (outliers) pull it strongly.
  • Median: the middle value when the data are in order. With an even number of values, average the two middle ones. Outliers barely affect it.
  • Mode: the value that appears most often. A set can have no mode, one mode or several. It is the only measure of center that works for categories, like favorite colors.

Worked example 1: all three

Quiz scores: 72, 85, 90, 85, 68, 95, 85.

  1. Mean: (72 + 85 + 90 + 85 + 68 + 95 + 85) / 7 = 580 / 7 ≈ 82.9.
  2. Median: in order, 68, 72, 85, 85, 85, 90, 95. The middle (4th) value is 85.
  3. Mode: 85 appears three times.

Choosing the right center

For data that are roughly symmetric, the mean and median are close and the mean is fine. For skewed data or data with outliers, the median is better. House prices are reported as medians because a few mansions would make the mean misleading. If the data in the example included a score of 5, the mean would drop to about 73.1 while the median would stay at 85.

Measures of spread

  • Range: maximum minus minimum. Simple but heavily affected by outliers. In the example, 95 − 68 = 27.
  • Interquartile range (IQR): Q3 − Q1, the spread of the middle 50% of the data. It resists outliers. A common rule flags any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR as an outlier.
  • Standard deviation: roughly the typical distance of values from the mean. Small means the data cluster tightly; large means they are spread out.

Worked example 2: standard deviation

Find the standard deviation of the data set 4, 6, 8, 10, 12, treating it as a whole population.

  1. Mean: 40 / 5 = 8.
  2. Deviations from the mean: −4, −2, 0, 2, 4.
  3. Square them: 16, 4, 0, 4, 16. Sum = 40.
  4. Divide by n for a population: 40 / 5 = 8. This is the variance.
  5. Take the square root: √8 ≈ 2.83.

If the data were a sample from a larger group, you would divide by n − 1 = 4 instead, giving a variance of 10 and a standard deviation of about 3.16. Dividing by n − 1 corrects for samples tending to underestimate spread.

Interpreting standard deviation

For data that follow a bell-shaped (normal) distribution, the empirical rule says about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. If test scores have a mean of 75 and a standard deviation of 8, about 95% of students scored between 59 and 91.

A z-score tells you how many standard deviations a value is from the mean: z = (x − mean) / SD. A score of 91 on that test has z = (91 − 75) / 8 = 2.

Common mistakes

  • Finding the median without sorting the data first.
  • Forgetting to take the square root at the end, which gives variance instead of standard deviation.
  • Mixing up sample (n − 1) and population (n) formulas.
  • Using the mean to describe heavily skewed data.

Practice questions

  1. Find the median of 3, 9, 4, 7, 1, 8.
  2. Which is more affected by an outlier, the mean or the median?
  3. Find the population standard deviation of 2, 4, 6.
  4. Heights have a mean of 165 cm and an SD of 7 cm. What is the z-score of 179 cm?

Answers: (1) Sorted: 1, 3, 4, 7, 8, 9, so the median is (4 + 7) / 2 = 5.5. (2) The mean. (3) Mean 4, squared deviations 4, 0, 4, variance 8/3, SD ≈ 1.63. (4) z = 2.

Last reviewed 2026-10-23 by the BrainBoost team.

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