12.3 Spread, Outliers & Statistical Displays

In Papers G–H and I–J, ICAS stops asking you simply to calculate and starts asking you to judge. What does an outlier do to the mean? Which display suits grouped continuous data? What does a scatterplot tell you about two variables? The official skills list names the effect of outliers on measures of location and spread, histograms, box plots and scatterplots — and the questions are interpretive, so a clear mental model matters more than fast arithmetic.

Outliers: Mean versus Median

An outlier is a value that lies far from the rest of the data. It affects the two main averages very differently:

• The mean uses every value, so an outlier drags it towards itself.
• The median only cares about position, so an outlier barely moves it.

Worked example. The masses of seven parcels are 2, 3, 3, 4, 4, 5, 29 kg.

• Mean = (2 + 3 + 3 + 4 + 4 + 5 + 29) ÷ 7 = 50 ÷ 7 ≈ 7.1 kg
• Median = the 4th value = 4 kg

The mean of about 7.1 kg describes none of the parcels well — six are 5 kg or less. The median of 4 kg is the honest summary. This is exactly the reasoning ICAS tests: if a question says one value is extreme and asks which measure is most affected, the answer is the mean; if it asks which best represents the data, the answer is usually the median. The range is also badly distorted by an outlier (29 − 2 = 27 here), because it depends on the two most extreme values.

Quartiles and the Interquartile Range

The range uses only two values, so statisticians prefer a spread measure based on the middle of the data. Quartiles split the ordered data into quarters:

• Q1 (lower quartile): the median of the lower half
• Q2: the median itself
• Q3 (upper quartile): the median of the upper half

The interquartile range, IQR = Q3 − Q1, is the spread of the middle 50% of the data. Because it ignores the top and bottom quarters, outliers do not affect it.

Worked example. For the eleven ordered values 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 21: the median (Q2) is the 6th value, 12. The lower half is 4, 6, 7, 9, 10, so Q1 = 7. The upper half is 13, 15, 16, 18, 21, so Q3 = 16. IQR = 16 − 7 = 9.

Box Plots

A box plot (box-and-whisker plot) draws the five-number summary — minimum, Q1, median, Q3, maximum — as a box with two whiskers:

• The box runs from Q1 to Q3, so it contains the middle 50% of the data.
• A line inside the box marks the median.
• The whiskers extend to the minimum and maximum.

Reading a box plot. If a box plot of test scores shows minimum 42, Q1 58, median 70, Q3 82, maximum 96, you can immediately state: the range is 96 − 42 = 54; the IQR is 82 − 58 = 24; half the students scored between 58 and 82; and a quarter scored 82 or more. Comparing two box plots is a classic ICAS task: the plot whose box sits further right has the higher typical value, and the plot with the longer box has the more spread-out middle half. Note a subtle point — the whiskers and box each hold a quarter of the data, so a short whisker does not mean 'fewer values', it means those values are tightly packed.

Histograms

A histogram displays grouped continuous data — measurements like heights, times or masses that fall anywhere in a range, grouped into intervals such as 10–≤20, 20–≤30. Unlike a column graph of categories, histogram columns touch: there are no gaps, because the intervals are continuous.

Key reading skills for ICAS:

• The modal class is the interval with the tallest column — you cannot name a single mode from grouped data, only the interval.
• Frequencies come from adding column heights; the total is the number of data values.
• With equal-width intervals, a column twice as tall represents twice the frequency.

A common trap is treating histogram intervals like separate categories and reading the horizontal position as a value rather than an interval.

Cumulative Frequency Histograms

Paper H adds the cumulative frequency histogram, which plots a running total rather than each interval's own count. Build it by adding each frequency to the sum of all those before it:

Three facts make these questions quick:

• The final cumulative value equals the total number of data values — 30 here. If your last bar does not match the total, you have made an addition slip.
• Cumulative frequency never decreases; the bars only climb or stay level.
• To answer 'how many took 30 minutes or less?', read the cumulative value at 30: 25. For 'how many took more than 30 minutes?', subtract from the total: 30 − 25 = 5.

The median can be estimated from a cumulative frequency graph by finding half the total (15 here) on the vertical axis, reading across to the curve and then down — which lands inside the 20–≤30 interval.

Back-to-Back Stem-and-Leaf Plots

Paper H also names back-to-back stem-and-leaf plots, which compare two groups against one shared stem. The first group's leaves run leftwards (so read them outwards from the stem, right to left) while the second group's run rightwards in the usual direction.

Key: 1 | 6 | 4 means 61 for Class A and 64 for Class B.

Class A holds 7 values (55, 58, 61, 63, 67, 72, 74) and Class B holds 8 (52, 56, 64, 65, 68, 69, 71, 73). Comparing them is the whole point of the display:

• Class A's median is its 4th value, 63. Class B's median is the mean of its 4th and 5th values, (65 + 68) ÷ 2 = 66.5, so Class B has the higher typical score.
• Class A's range is 74 − 55 = 19, narrower than Class B's 73 − 52 = 21, so Class A is the more consistent group.

Reversing the left-hand leaves is the most common slip — always read them outwards from the stem.

Scatterplots and Correlation

A scatterplot places one point per pair of measurements — for example, hours of practice versus score — to show whether two variables are related.

• Positive correlation: points rise from left to right (more practice, higher scores).
• Negative correlation: points fall from left to right (more absences, lower marks).
• No correlation: points scatter with no pattern.

Correlation can also be described as strong or weak depending on how tightly the points hug an imaginary straight line. Two ICAS cautions. First, correlation is not causation: ice-cream sales and drowning incidents are correlated because both rise in hot weather, not because one causes the other. Second, one odd point sitting far from the pattern is an outlier, and ICAS may ask you to identify it.

Choosing and Comparing Displays

Match the display to the data:

Worked interpretation. Two box plots compare reaction times for two games. Game A: median 310 ms, IQR 40 ms. Game B: median 280 ms, IQR 90 ms. Which game gives faster typical responses? Game B — its median is lower (lower time is faster). Which gives more consistent responses? Game A — its smaller IQR means the middle half of its times are closer together. Being able to say both things, typical level and consistency, is precisely what Papers I–J reward.

Common Traps

• Claiming an outlier 'changes the median a lot' — it usually changes it by one position at most.
• Reading the ends of a box plot's whiskers as Q1 and Q3; they are the minimum and maximum.
• Saying grouped data has a single mode rather than a modal class.
• Treating strong correlation as proof that one variable causes the other.