Statistics: Data, Averages & Diagrams

Types of Data

Statistics begins with knowing what kind of data you are handling, because it decides which averages and diagrams are sensible.

Qualitative (categorical) data describes qualities — colours, names, types. It cannot be averaged numerically.
Quantitative data is numerical. It splits into discrete data (countable values such as the number of goals: $0, 1, 2, \dots$) and continuous data (measured on a scale such as height or time, which can take any value in a range).

Key terms Primary data is collected first-hand by you; secondary data is taken from an existing source.

Population is the whole group of interest; a sample is a smaller part chosen to represent it. A good sample is large and unbiased.

Mean, Median, Mode and Range

For a simple list of values:

Mean $= \dfrac{\text{sum of values}}{\text{number of values}}$ — uses every value, but is distorted by outliers.
Median is the middle value when the data is in order. For $n$ values it lies at position $\frac{n+1}{2}$.

Worked example For $4, 7, 7, 9, 13$: mean $=\frac{40}{5}=8$, median $=7$ (3rd value), mode $=7$, range $=13-4=9$.

Mean from a Frequency Table

When data is grouped by frequency, multiply each value $x$ by its frequency $f$, total the products, then divide by the total frequency.

$$\bar{x}=\frac{\sum fx}{\sum f}$$

Mean $=\dfrac{25}{20}=1.25$ goals. The mode is $1$ (highest frequency). The median is the $\frac{20+1}{2}=10.5$th value, which falls in the "1" group, so the median is $1$.

Estimated Mean from Grouped Data

With grouped continuous data you do not know exact values, so use the midpoint of each class as a best estimate of $x$.

Worked example Times $t$ (minutes) for $40$ runners:

| Time (min) | Freq $f$ | Midpoint $x$ | $fx$ | |---|---|---|---| | $0<t\le10$ | 6 | 5 | 30 | | $10<t\le20$ | 14 | 15 | 210 | | $20<t\le30$ | 13 | 25 | 325 | | $30<t\le40$ | 7 | 35 | 245 |

Watch out It is only an estimate — say "estimated mean" and never give exact-looking accuracy. Use midpoints, not class boundaries.

Statistical Diagrams

Bar charts show frequencies of categories using bars of equal width with gaps; the height is the frequency.
Pictograms use a symbol to represent a number of items; always read the key.
Pie charts show proportions. Each category's angle $=\dfrac{\text{frequency}}{\text{total}}\times360^\circ$. For a total of $20$ people, one person $=18^\circ$.

Exam tip In a pie chart question, to go back from an angle to a frequency, divide the angle by $360^\circ$ and multiply by the total.

Scatter Graphs and Correlation

A scatter graph plots paired data to reveal a relationship.

Positive correlation: as one variable rises, so does the other.
Negative correlation: as one rises, the other falls.
No correlation: no clear pattern.

A line of best fit is a straight line following the trend with roughly equal points either side, passing through the mean point $(\bar{x},\bar{y})$. Use it to estimate values — interpolation (within the data) is reliable; extrapolation (beyond it) is risky.

Watch out Correlation does not prove causation. Two things may rise together because of a third hidden factor.

Cumulative Frequency

Cumulative frequency is a running total of frequencies. Plot it against the upper class boundary of each group, then join the points with a smooth curve.

To find the median, read across from $\frac{n}{2}$ (for a curve, use $\frac{n}{2}$, not $\frac{n+1}{2}$). The lower quartile $Q_1$ is read at $\frac{n}{4}$ and the upper quartile $Q_3$ at $\frac{3n}{4}$.

$$\text{Interquartile range (IQR)} = Q_3 - Q_1$$

Worked example For $n=40$: median at $\frac{40}{2}=20 \Rightarrow \approx 20$ min; $Q_1$ at $10 \Rightarrow \approx 13$ min; $Q_3$ at $30 \Rightarrow \approx 27$ min.

Box Plots

A box plot (box-and-whisker) summarises five numbers: minimum, $Q_1$, median, $Q_3$, maximum. The box spans the IQR; the line inside is the median; whiskers reach the extremes.

Box plots make it easy to compare two distributions: compare medians for average and IQRs (box widths) for consistency.

Histograms with Unequal Class Widths

When class widths differ, bar heights must show frequency density, not frequency — otherwise wide classes look misleadingly large. The area of each bar equals the frequency.

$$\text{frequency density} = \frac{\text{frequency}}{\text{class width}}$$

Worked example | Mass $m$ (kg) | Freq | Width | Freq density | |---|---|---|---| | $0<m\le10$ | 8 | 10 | 0.8 | | $10<m\le20$ | 18 | 10 | 1.8 | | $20<m\le40$ | 24 | 20 | 1.2 | | $40<m\le70$ | 9 | 30 | 0.3 |

Exam tip The golden rule is frequency = area = frequency density $\times$ class width. If a question gives you density and width, multiply; if it gives frequency and width, divide. Bars in a histogram have no gaps.