Mathematics · Combinatorics, probability and statistics · Grade 8

Statistics: mean, median, σ

What is the “typical” height of a class? By how much does one very tall student shift the average?

The situation

The heights of 25 students in a class were measured, and the steps they walked each day for a week were recorded. What is the “typical” height of the class? By how much does one very tall student shift the mean height? How spread out are the heights? Is height related to shoe size, and what about height and steps? We turn 25 numbers from a table into a diagram, and from the diagram into one or two meaningful numbers.

Topic

Data analysis: tables and charts, the mean, mode, median, standard deviation, the relation between two quantities. Heights of the class and weekly step counts: the histogram, the mean, the median, spread, the scatter plot

Key ideas

Histogram
Mean and median
Standard deviation

Formulas

\overline{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n}

The mean (arithmetic mean) is the sum of all the values divided by their number — the height we get when the columns are “levelled out”.

M e = x_{\frac{n + 1}{2}} (n odd), M e = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2} + 1}}{2} (n even)

The median is the middle value of the sorted list; with an even number of values, the mean of the middle two.

σ = \sqrt{\frac{{(x_{1} - \overline{x})}^{2} + \dots + {(x_{n} - \overline{x})}^{2}}{n}}

Standard deviation: the square root of the mean of the squared deviations — how spread out the data are around the mean.

r = \frac{\sum (x_{i} - \overline{x}) (y_{i} - \overline{y})}{\sqrt{\sum {(x_{i} - \overline{x})}^{2} \cdot \sum {(y_{i} - \overline{y})}^{2}}}

Correlation coefficient $- 1 \leq r \leq 1$ : near 1 — strong positive, near $- 1$ — strong negative, around 0 — no visible relationship.

w_{i} = \frac{n_{i}}{n}, {\overline{x}}_{grp} \approx \frac{\sum m_{i} n_{i}}{n}

Relative frequency is the share of the values in a group; for the grouped mean, the class midpoints $m_{i}$ are multiplied by the frequencies and added up.

Theory

Displaying data

25 numbers in a table are hard to take in. A bar chart gives a bar to every number, but bars in no particular order do not say much either. A histogram splits the values into groups (intervals) of equal width and shows how many values each group contains — the frequency: the 160–165 cm group has the most students. A pie chart shows shares: 9 out of 25 is 36 %. In the first step, change the group width — how does the picture change?

The mean

Six students walk 4, 6, 7, 9, 5 and 8 thousand steps a day. If we pool all the steps and share them out equally, each gets $\frac{39}{6} = 6.5$ thousand — this is the mean. The columns are “levelled”: the tall ones give to the short ones, and the total does not change. That is why the mean depends on every value: raise one, and the mean rises too — in the second step, drag the top of a column.

Mode and median

The mode is the most frequent value (in the class, 160 cm — 4 students). The median is the middle of the sorted list: half the values are below it and half above; with 8 values it is the mean of the 4th and 5th. A single outlier (one student walked 40 thousand steps) shifts the mean strongly but hardly affects the median — the middle position does not change. That is why the median is more reliable for data with outliers, such as salaries or house prices.

Deviation and standard deviation

Two classes can have the same mean height, yet in one almost everyone is the same height and in the other the heights are very spread out. The deviations $x_{i} - \overline{x}$ show the spread, but their sum is always 0 (the positives cancel the negatives). So we square the deviations, take their mean (this is the variance) and take the root to return to the original unit: $σ = \sqrt{\frac{\sum {(x_{i} - \overline{x})}^{2}}{n}}$ . If we increase every deviation by a factor of 2, $σ$ also increases by a factor of 2 — in the fourth step there is a spread slider.

The relationship between two kinds of data

When each student has two numbers (height and shoe size), we plot them as points on a scatter plot. If the points run upwards along a line, there is a positive correlation (taller — bigger shoes); downwards, a negative one (more training — less time for 100 m); if they form a shapeless cloud, there is no visible relationship (height and steps). The strength of the relationship is measured by the correlation coefficient $r$ , and the straight line closest to the points is the trend line. Correlation does not mean causation: ice-cream sales and swimming accidents rise together because both depend on summer.

Frequencies, samples and the model

The relative frequency $w_{i} = n_{i} / n$ is the share of a group (they add up to 1); the cumulative frequency is how many values have accumulated up to and including that group. If only the groups are known, the mean is approximate: class midpoints times frequencies give 161.5 cm, close to the true 160.8. The whole class is the population, and a few students taken from it are a sample: the sample mean is close to the population mean but not equal to it, and the larger the sample, the closer. In this experiment $σ$ is calculated by the school formula (dividing by $n$ ); in scientific work one often divides by $n - 1$ — the result is slightly larger.

Did you know?

The word “statistics” was introduced in 1749 by the German scholar Gottfried Achenwall in the sense of “the science of the state” (Latin status — state, condition).
In 1858 Florence Nightingale showed Parliament, with “rose” (polar area) diagrams of deaths in the hospitals of the Crimean War, that soldiers were dying more from disease than in battle.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

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