Statistics

Averages and box plots

Move the data. Follow centre and spread.

Predict, test and explain

Step 1 of 3

Predict

For 1, 3, 4, 5, 6, 7, 9, replace 9 with 20. What happens?

xˉ=357=5\bar{x}=\frac{35}{7}=5
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mean 5\text{mean }5
median 5\text{median }5

Scroll the diagram sideways to see every label.

median=5\text{median}=5range=8\text{range}=8

InvestigateMove one value far away. Which average changes most?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Find the mean and median of 2, 3, 4, 5 and 16.

2,  3,  4,  5,  162,\;3,\;4,\;5,\;16

Hint

Divide the total by five; the ordered middle value gives the median.

Reveal answer

The mean is 6 and the median is 4. The large final value raises the mean.

xˉ=305=6,median=4\bar{x}=\frac{30}{5}=6,\quad \text{median}=4

Build an explanation

Hint 1

Order the data before finding the middle value.

Hint 2

Mean uses every value; median uses the middle position.

Hint 3

For seven values, exclude the middle value before finding the medians of the lower and upper halves.

Worked example

This example uses fixed values, separate from the diagram controls.

For 1, 3, 4, 5, 6, 7, 9, find the mean, median and quartiles.

  1. The total is 35, so the mean is 35/7 = 5.
  2. The fourth value is 5, so the median is 5.
  3. The lower half is 1, 3, 4 and the upper half is 6, 7, 9.

Answer: Mean 5; median 5; Q₁ = 3; Q₃ = 7.

Connect this idea

Watch out: A box plot shows position and spread; its box covers the middle half of the data.