Unit 4: Statistics — Measures of Dispersion

• Mean x̄ = (sum of the values) ÷ (their number); for a frequency table x̄ = Σfx ÷ Σf.
• Median: the middle value after sorting the data; with an even count it is the mean of the two central values.
• Mode: the most frequent value; a set may have no mode or more than one.
• All three describe the centre only — they say nothing about how the values are spread.

• Two data sets can share the same mean and still behave very differently: {10, 10, 10} and {1, 10, 19} both have mean 10, but the second is far more spread out.
• A measure of dispersion quantifies how scattered the values are around the centre, which is what consistency, risk and stability mean in real decisions.
• Examples of decisions that need dispersion: choosing the more consistent player, the machine with steadier output, or the investment with less variability.

• Range = greatest value − least value; it uses only two numbers, so it is easy but sensitive to extreme values.
• From a frequency table read the largest and smallest value with a non-zero frequency, not the largest frequency.
• From a stem-and-leaf plot the extremes are the first and last leaves with their stems.
• For a continuous class interval, the range is usually taken as the difference between the upper boundary of the last class and the lower boundary of the first class.
• The range carries the same unit as the data (it is not squared).

• Sort the data, then split it: Q1 is the median of the lower half, Q2 is the median of the whole set, Q3 the median of the upper half.
• Interquartile range (IQR) = Q3 − Q1 — the spread of the middle 50% of the data, and unlike the range it is not distorted by extreme values.
• When the count is odd, the common school convention excludes the middle value from both halves.
• Half of the data always lies inside the interval [Q1, Q3], which is why the IQR is used to judge consistency of the bulk of the observations.

Worked example

• Data 4, 6, 7, 9, 14 sorted: median Q2 = 7, lower half {4, 6} → Q1 = 5, upper half {9, 14} → Q3 = 11.5, so IQR = 6.5 and range = 10.

• A box plot is drawn from exactly five numbers: the minimum, Q1, the median, Q3 and the maximum.
• The box spans Q1 to Q3 (so its length is the IQR), the line inside the box is the median, and the whiskers reach the least and greatest values.
• Reading it: a longer box means more spread in the middle 50%; a whisker much longer than the other signals a possible outlier; the side on which the median line sits shows where the data concentrate.
• Two box plots on the same axis compare the two distributions at a glance — position (median) and spread (box length) together.

• The deviation of a value is x − x̄; the deviations always sum to zero, which is why they are squared.
• Variance σ² = Σ(x − x̄)² / n and standard deviation σ = √σ² — the SD returns to the original unit of the data, so it is the more usable number.
• σ = 0 only when all the values are equal; the larger σ, the more scattered the data.
• Shortcut formula: σ² = (Σx²/n) − (x̄)² — often faster and less rounded.

Worked example

• For 4, 6, 7, 9, 14: x̄ = 8; the squared deviations are 16, 4, 1, 1, 36 with sum 58, so σ² = 58/5 = 11.6 and σ ≈ 3.41.

• Work with columns x, f, fx and fx²: n = Σf, x̄ = Σfx/Σf, and σ² = Σfx²/Σf − (x̄)².
• Each squared deviation is weighted by its frequency: σ² = Σf(x − x̄)² / Σf — the two forms give the same result.
• Build the table fully before substituting; a single wrong fx² entry moves the final answer.

• Replace each class by its midpoint x = (lower + upper)/2 and treat the table as a frequency distribution with those midpoints.
• Then apply σ² = Σfx²/Σf − (Σfx/Σf)²; the answer is an estimate because the real values inside a class are unknown.
• Use class boundaries, not the printed limits, when computing midpoints of continuous classes such as 10–19, 20–29.
• Quartiles of grouped data are read from the cumulative-frequency curve or located in the class containing the required position.

• Enter statistics mode (usually MODE → STAT → 1-VAR), type each value with its frequency, then read x̄, σn (population SD) and σn−1 (sample SD) from the menu.
• The syllabus uses σn (divide by n); do not copy σn−1 (divide by n − 1) by mistake.
• Frequencies are entered through the frequency column; forgetting it treats each value as appearing once.
• Keep the unrounded calculator value until the final step, then round to the required number of decimal places.

• Compare centre with the mean (or median) and consistency with the standard deviation or the IQR: the smaller the dispersion, the more consistent the performance.
• Standard deviation is in the data's unit, so it is directly comparable only when the two means are close; otherwise use the coefficient of variation CV = (σ / x̄) × 100% and pick the smaller percentage.
• Typical wording: "which player is more consistent?" — the answer is the one with the smaller SD/IQR, even if his mean score is lower.
• Justify with numbers in the answer sentence; a bare name earns no mark.

• Adding a constant c to every value: the mean and median increase by c, quartiles increase by c, but the range, IQR and standard deviation do not change — the spread is only shifted.
• Multiplying every value by k: mean, median, quartiles, range, IQR and SD are all multiplied by |k|, and the variance is multiplied by k².
• Change of unit (e.g. metres to centimetres) is a multiplication by 100, so SD ×100 and variance ×10 000 while the coefficient of variation stays unchanged.

• An outlier is a value far removed from the rest; it drags the mean and inflates the standard deviation and the range, but barely affects the median and the IQR.
• A common test flags a value as an outlier if it lies below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
• When an outlier is present (or the distribution is heavily skewed), report the median with the IQR; otherwise the mean with the standard deviation is the better pair.
• Never delete an outlier silently: either justify its removal (measurement error) or keep it and switch to the robust measures.