Statistics

Data is collected from people — a sample drawn from a population.

To organise statistical 统计 data 数据, use:

• a tally table 计数表 — make a mark for each value, then count.
• a two-way table 双向表 — sorts data by two features at once (for example, boys/girls against walk/bus).

When you read a diagram, only draw conclusions the data really supports.

Three averages describe a typical value of the data, and the range shows how spread out it is. Each is used for a different purpose:

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

• mean 平均数 $= \dfrac{\text{sum of all values}}{\text{how many values}}$.
• median 中位数 $=$ the middle value when the data is put in order.
• mode 众数 $=$ the value that appears most often.
• range 极差 $=$ largest value $-$ smallest value (it shows how spread out the data is).

Worked example. Find the mean, median, mode and range of $4, 7, 7, 2, 5$.

Order the data: $2, 4, 5, 7, 7$.

When data is listed with its frequency 频数 (how many times each value occurs), the mean is

Worked example. Values $1, 2, 3$ occur with frequencies $4, 5, 1$. Find the mean.

Grouped data (Extended)

For grouped data 分组数据, you cannot find the exact mean, so estimate it using the midpoint of each group as the value. The modal class 众数组 is simply the group with the highest frequency.

When data is in order, the quartiles 四分位数 cut it into four equal parts. The lower quartile (LQ) is one quarter of the way up; the upper quartile (UQ) is three quarters of the way up. The interquartile range 四分位距 measures the spread of the middle half:

The interquartile range is useful because, unlike the range, it ignores extreme values.

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

A Galton board shows how data piles up into a distribution.

A pictogram uses a symbol to stand for a number of items

A bar chart shows the frequency of each group

Worked example (pie chart). Out of $120$ people, $30$ chose tea. Find the angle of the tea slice.

Each slice is its fraction of $360^\circ$; tea is $\tfrac{30}{120}\times360^\circ=90^\circ$.

A scatter diagram 散点图 plots pairs of values as points to show whether two things are linked. The link is the correlation 相关性:

• positive correlation 正相关 — as one goes up, the other goes up.
• negative correlation 负相关 — as one goes up, the other goes down.
• zero correlation 零相关 — no clear link.

Positive correlation rises together; negative falls as the other rises; zero shows no clear link.

If there is correlation, draw a line of best fit 最佳拟合线: one straight ruled line through the middle of the points, with about the same number of points on each side. Use it to predict values.

A line of best fit is one straight ruled line through the middle of the points; use it to predict.

The cumulative frequency 累积频数 is a running total of the frequencies. Plot it against the upper end of each class and join the points with a smooth curve.

From the curve you can read the median (at half the total), the quartiles (at one quarter and three quarters), and any percentile 百分位数 (for example, the $90$th percentile is at $90\%$ of the total).

Read the median at half the total, and the quartiles at one quarter and three quarters: across to the curve, then down.

A histogram 直方图 looks like a bar chart, but the bars can have different widths and the area of each bar (not its height) shows the frequency. The vertical axis is the frequency density 频数密度:

With unequal class widths the bar AREA (not its height) is the frequency, so the axis is frequency density.

Worked example. A class has class width 组距 $10$ and frequency $25$. Find the frequency density.

• Know the three averages: mean (add up ÷ how many), median (the middle value when in order), mode (the most common). The range is highest − lowest.
• From a frequency table the mean is $\frac{\sum fx}{\sum f}$ — divide by the total frequency, not the number of rows.
• On a scatter diagram, describe the correlation as positive, negative or none, and draw the line of best fit through the mean point.
• For a histogram with unequal class widths, the height is the frequency density (frequency ÷ class width), not the frequency.