Mean, Median, Mode, and Range

Quantitative data

The data you summarize in this lesson is quantitative data: numerical, not categorical. Quantitative data splits into two kinds (the same distinction from Lesson 32, applied here to data instead of to a function):

• Discrete data are countable values: number of students absent, number of goals scored, number of broken windows. The values are integers (or at least separated values); there's no meaningful "halfway between two counts."
• Continuous data are measurable values: height, weight, temperature, time elapsed. The values can land anywhere on a continuous scale, and "halfway between two measurements" is a real possibility.

Data collected from a sample can be summarized by using statistics.

The four statistics in this lesson (mean, median, mode, range) work on quantitative data, both discrete and continuous. They don't work on categorical data (favourite colour, type of animal, name of city); a different toolkit handles those.

Mean: the arithmetic average

The mean is the value you get by adding all the data and dividing by how many there are. The notation is x̄ ("x-bar"):

For the data set 8, 6, 10, 4, 12: the sum is 40, there are 5 values, so the mean is 40 / 5 = 8. The mean is what most people mean by "average."

The mean takes every value into account equally. That makes it sensitive to extreme values: one very high or very low value drags the mean toward it. That sensitivity is sometimes useful (it catches the influence of an outlier) and sometimes a problem (it misrepresents the bulk of the data).

Median: the middle

The median is the middle value of the sorted data. To find it:

1. Sort the values from smallest to largest.
2. Pick the middle one.

For an odd count, there's a single middle. For an even count, there are two middles, and the median is the mean of those two.

Two worked examples:

• Odd count: 3, 7, 12, 4, 9. Sort: 3, 4, 7, 9, 12. The middle is the third value (out of five), which is 7. Median = 7.
• Even count: 2, 4, 7, 11. Sort: 2, 4, 7, 11. The two middles are 4 and 7. Median = (4 + 7) / 2 = 5.5.

The median takes only the rank of the values into account, not their exact magnitude. That makes it robust to outliers: one extreme value can't drag the median far, because moving it past its rank doesn't change which value is in the middle.

Mode: the most common value

The mode is the value that appears most often. A data set can have:

• One mode (unimodal): the value most repeated.
• Multiple modes (multimodal): two or more values tied for most repetitions.
• No mode: every value appears the same number of times (often once each).

For 5, 5, 7, 9, 12, the mode is 5: it appears twice, every other value once. For 5, 5, 7, 7, 12, the modes are 5 and 7 (both appear twice). For 5, 7, 9, 12, there's no mode; each value appears once.

The mode is especially useful for discrete data that takes a small number of distinct values. For continuous measurements (heights in centimetres, temperatures in degrees), the mode is usually not very informative because two measurements rarely come out identical.

Range: the spread

The range is the simplest measure of spread: the difference between the largest and smallest values.

For 8, 6, 10, 4, 12: max = 12, min = 4, range = 8. The range tells you the width of the data set's reach, but nothing about how the values are distributed in between.

See it: drag the dots

The widget below has seven values plotted as dots on a horizontal axis (a dot plot). Each dot represents one day's high temperature in Edmonton. Use the ◀ / ▶ buttons to slide each dot left or right, in one-degree steps. The four statistics (mean, median, mode, range) update live.

Try these specific experiments:

1. With the default Edmonton week, note where the mean and median sit. They should be close.
2. Switch to the "Same week + outlier" preset (one day at −25 °C). The MEAN drops sharply, pulled toward the outlier, but the MEDIAN barely moves. Same data minus one value; the two statistics react very differently.
3. Slide several dots to repeat at the same value. The mode is whatever's most repeated. Slide them all to be different and the mode disappears.

Slide the values to build intuition for which statistic is robust and which is sensitive.

Outliers: mean shifts, median holds

Mean vs. median, in the presence of an outlier (one value that sits far outside the rest):

• The mean uses every value's magnitude, so an outlier far from the bulk pulls the mean toward it. A −25 °C day in a week of mostly mild temperatures drops the mean by several degrees.
• The median uses only ranks, so an outlier can move from far-out to slightly-far-out without changing the middle. The median stays put.

Here's the whole effect, worked through one cold snap. Take a mild Edmonton week, in °C:

• Before: −1, 0, 1, 2, 2, 3, 7. Sum 14, so the mean is 14 ÷ 7 = 2. Sorted, the middle (4th) value is 2: median 2. Mode 2, range 7 − (−1) = 8.
• Now a cold snap replaces the 7 °C day with −25 °C: −25, −1, 0, 1, 2, 2, 3. Sum −18, so the mean is −18 ÷ 7 ≈ −2.6. Sorted, the middle value is 1: median 1. Mode still 2, range now 3 − (−25) = 28.

Score the damage: the mean fell 4.6 degrees — from 2 to −2.6 — dragged the full distance by one value. The median slipped one degree (2 to 1), because the cold day only shuffled who sits in the middle seat. The range exploded from 8 to 28, which is exactly its job: range reports extremes.

That's why news reports on income, house prices, and other heavily-skewed data usually report the median rather than the mean. Mean income is dragged up by a few very high earners; median income tells you what a typical person makes. The right statistic depends on what the data looks like and what question you're asking.

Drawing conclusions about the population

The sample's statistics estimate the population's statistics. If you sample 50 Grade 7 students and find a median screen time of 2.5 hours per day with a mean of 3.1 hours per day, the typical Grade 7 student probably uses around 2.5–3 hours a day, and the gap between mean and median tells you there are some high-screen-time outliers pulling the mean up.

Drawing the conclusion explicitly is part of the work. The numbers themselves don't say what they mean; you have to interpret them in the context of the population the sample came from.

Where it shows up in real life

Hourly temperatures during a single Edmonton week are the lesson's running example. Recording the high once per day gives you seven discrete data values. The mean tells you the week's "average" temperature; the median tells you the typical day; the mode (if any) tells you which temperature recurred; the range tells you the week's variability. Together, the four describe the week's weather.

Mountain Pine Beetle infestation rates across multiple sample plots are summarized the same way. A forestry team measures the infested proportion in each of, say, 30 sample plots. The mean infested rate across plots is their estimate for the stand; the median tells them the typical plot; the range tells them whether the infestation is uniform or patchy.

A class's quiz scores: mean tells you class performance, median tells you the typical student's score, mode tells you which score showed up most often, range tells you the spread. Teachers use all four routinely.

Worksheet

These aren't graded. Get them right, get them wrong. The goal is fluency with each statistic: compute it correctly, then interpret what it says about the data.

Going further

This is the end of the Statistics strand for Grade 7. Together with Populations and Samples, you now have the full Grade 7 toolkit: identify a population, draw a representative sample, and summarize the sample with four statistics that estimate the population's.

In Grade 8 and 9, the toolkit expands. You'll meet measures of spread beyond range (variance, standard deviation) that use every value's distance from the mean rather than just the extremes. You'll learn how to choose measures of central tendency more carefully (mean for symmetric data, median for skewed data, mode for categorical-like data) and how to recognize which statistic each newspaper or government report is quietly using. The discrete-vs-continuous distinction from this lesson (and from Lesson 32 on Domain and Range) continues to shape what techniques apply.

The four statistics in this lesson are the start of a long conversation about how to summarize data honestly. Get fluent with them, and what comes next will fall into place.