12.8 Summary
Key Points
- The mean is a balance point: the total distance of values on its left equals the total distance on its right, and this balance point is unique.
- Inserting a value above the mean raises it; below, lowers it; equal to it, leaves it unchanged. Deleting values reverses these effects. The median shifts only by which side a new value lands on, never minding how far.
- Shifting every value by c changes the mean by c; multiplying every value by c multiplies the mean by c.
- With a frequency table, the mean is \dfrac{\sum (\text{value}\times\text{frequency})}{\sum \text{frequency}}, and the median follows from cumulative frequencies. The mode is the most frequent value.
- A few extreme values (outliers) can drag the mean far from the bulk of the data; the median resists them, so it is often the fairer “typical value.”
- A known mean lets you recover a missing value, since \text{sum} = \text{mean}\times\text{count}, and lets you combine groups by pooling totals.
- Spreadsheets use cells (e.g.
E5), ranges (e.g.B3:G3) and functions (=SUM,=AVERAGE) to summarise data fast. - Line graphs show change over time; steeper segments mean faster change. They beat clustered bars when there are many points. Infographics and activity strips pack rich data into one readable picture.
- Reading data well doesn’t just answer questions, it breeds new ones worth chasing.