Averages and range

What's "normal"?

• The average salary in a company is 50 000 — but most people earn far less, because one CEO earns millions.
• Which average you choose changes the story. Mean 平均数, median 中位数, and mode 众数 each tell a different truth.

The three averages

• Mean $= \dfrac{\text{sum of values}}{\text{how many}}$.
• Median $=$ the middle value when the data is ordered.
• Mode $=$ the most common value.

A box-and-whisker plot shows the five-number summary; the box length is the interquartile range 值域

The range

• Range $=$ largest $-$ smallest. It measures spread.
• Range of $4, 7, 7, 2, 5$: $7 - 2 = 5$.

Mean, median, mode and range of a small data set

Mean from a frequency table

• When values come with frequencies:

• Values $1, 2, 3$ with frequencies $4, 5, 1$: $\;\dfrac{1(4) + 2(5) + 3(1)}{10} = \dfrac{17}{10} = 1.7$.

Which average to use?

• Mean: uses all the data, but affected by outliers.
• Median: usually less affected by extreme values than the mean — good for skewed data (like salaries).
• Mode: useful for categorical data (most popular colour, most common shoe size).

Ordering data on a number line makes the median obvious — it's the value in the middle.

Even counts and frequency-table medians

• For $2,5,7,10$, the middle pair is 5 and 7, so median $M=(5+7)/2=6$; do not select only one middle value.
• Values 1,2,3 with frequencies 2,3,1 make six observations. Positions 3 and 4 are both 2, so median 2. Frequency counts determine the middle positions, not the number of table rows.

You've got it

Write from a mean of 6

• The values are 3, 3, 8 and 10.
• Find the mean.