Statistics

Averages & Range

An average is a single number standing in for a whole data set. Three are used at IGCSE, each a different sense of "typical"; the range measures how spread out the values are.

The three averages and the range

The mean shares the total equally: mean = (sum of values) ÷ (number of values). The median is the middle value once the data is ordered. The mode occurs most often. The range is the gap between the extremes, highest − lowest.

Worked example: a team scores 1, 3, 0, 2, 3, 4, 2, 3, 1 goals across nine matches. Find the mean, median, mode and range. Step 1: order the data — 0, 1, 1, 2, 2, 3, 3, 3, 4. Step 2: mean = 19 ÷ 9 = 2.11 (3 s.f.). Step 3: median = 5th value = 2; mode = 3 (three times); range = 4 − 0 = 4 — each answers a different question. Mean 2.11, median 2, mode 3, range 4.

The mean from a frequency table

When values repeat, a frequency table is quicker. Multiply each value x by its frequency f, total the products, and divide by the total frequency.

Worked example: in a class of 40, the number of pets owned is 0 (×8), 1 (×14), 2 (×10), 3 (×6), 4 (×2). Find the mean. Step 1: Σf = 8 + 14 + 10 + 6 + 2 = 40. Step 2: Σfx = 0 + 14 + 20 + 18 + 8 = 60. Step 3: mean = 60 ÷ 40 = 1.5 pets per student.

Grouped Data & Spread

Once data is grouped into classes, the individual values are gone: we can only estimate the mean, and we measure spread with the interquartile range, which ignores the extremes.

Estimating the mean of grouped data

We assume every value sits at its class midpoint, then work out estimated mean = Σfx ÷ Σf as before, where x is the midpoint of each class. The modal class is the interval with the greatest frequency.

Worked example: the mass of 60 apples is grouped as 100<m≤120 (f = 8), 120<m≤140 (f = 20), 140<m≤160 (f = 22), 160<m≤180 (f = 10). Estimate the mean and state the modal class. Step 1: midpoints 110, 130, 150, 170. Step 2: Σfx = 880 + 2600 + 3300 + 1700 = 8480. Step 3: estimated mean = 8480 ÷ 60 = 141.3 g (1 d.p.). Step 4: the highest frequency is 22, so the modal class is 140<m≤160.

Quartiles and the interquartile range

For a list, find the median, then Q₁ is the middle of the lower half and Q₃ the middle of the upper half. The IQR, Q₃ − Q₁, measures the spread of the central half.

Worked example: eleven ordered test scores are 4, 6, 7, 7, 8, 9, 11, 12, 12, 14, 15. Find the quartiles and the IQR. Step 1: median Q₂ = 6th value = 9. Step 2: lower half 4, 6, 7, 7, 8 → Q₁ = 7; upper half 11, 12, 12, 14, 15 → Q₃ = 12. Step 3: IQR = 12 − 7 = 5. Q₁ = 7, Q₃ = 12, IQR = 5.

Charts & Diagrams

The right diagram makes a pattern obvious at a glance. Each has a job: bar charts compare categories, pie charts show proportions of a whole, and stem-and-leaf diagrams keep every value while revealing the shape of the data.

Pie charts: angles from frequencies

A full circle is 360° and represents the whole data set. Each category takes a slice whose angle is its share of the total: sector angle = (frequency ÷ total) × 360°, and the angles must sum to 360°. Composite (stacked) and dual (side-by-side) bar charts do the same comparison job for two sub-groups at once.

Worked example: of 60 students, 24 chose football. Find the angle of the football sector. Step 1: fraction = 24 ÷ 60. Step 2: angle = (24 ÷ 60) × 360° = 144°. Football sector = 144°.

Scatter & Correlation

A scatter diagram plots two variables against each other, one point per item, to reveal whether they are linked. If the points trend along a line, a line of best fit turns that trend into predictions.

Reading correlation

Points rising left to right show positive correlation; falling points show negative correlation; a shapeless cloud shows zero correlation. The tighter the points cluster to a line, the stronger the correlation. Plot points clearly, as small crosses.

Worked example: a student revised for 5 hours but was absent for the test. Use the line of best fit to estimate their score. Step 1: find 5 hours on the horizontal axis and read up to the line. Step 2: read across to the score axis — about 62 marks. Step 3: this is an estimate — 5 hours lies inside the data, so it is reasonable. Estimated score ≈ 62.

Cumulative Frequency

When data is grouped, the median and quartiles cannot be found exactly — but a cumulative frequency curve lets us estimate them by reading values straight off the graph. The same grouped table also builds a histogram, covered next.

Building and reading the curve

Form a running total of the frequencies, then plot each cumulative total against the upper boundary of its class and join the points with a smooth curve. To estimate the median of n values, go to n ÷ 2 on the cumulative axis, across to the curve and down to the value.

Worked example: for the 80 runners above, estimate the median and the interquartile range from the curve. Step 1: median at 40 — read off ≈ 57 s. Step 2: Q₁ at 20 → 50 s; Q₃ at 60 → ≈ 65 s. Step 3: IQR = 65 − 50 = 15 s. Median ≈ 57 s, IQR ≈ 15 s.

Histograms

When class intervals have unequal widths, a bar chart of raw frequency misleads — a wide class looks bigger simply because it is wide. A histogram fixes this by making area, not height, represent frequency.

Frequency density

Divide each frequency by its class width to get the frequency density, and use that as the bar height: frequency density = frequency ÷ class width, so frequency = frequency density × class width. Now a bar's area equals its frequency, so classes of different widths are compared honestly.

Worked example: in the class 20<h≤40, the frequency is 40. Find the frequency density. Step 1: class width = 40 − 20 = 20. Step 2: frequency density = 40 ÷ 20 = 2.0. Step 3: a wide class with a modest density — area still records all 40 values.