Dispersion and Standard Scores
FoundationThe mean only captures central tendency; dispersion describes how spread out data are, with range, inter-quartile range, variance and standard deviation each playing a part — and the standard score converts raw marks into relative positions so results from different subjects can be compared fairly.
- Dispersion describes the spread of data, paired with central tendency as the two key features
- Range = maximum − minimum; inter-quartile range = Q3 − Q1, robust against extreme values
- Variance = the average squared deviation; standard deviation = its positive square root, sharing the data's unit
文體小知識：數學概念說明文——講解離差的概念，逐一介紹分佈域、四分位數間距、方差與標準差，並以標準分把不同組別的表現換算成可公平比較的相對位置。
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1Two classes can score exactly the same average yet perform very differently: one clustered near the mean, the other split at the extremes. To describe a data set we need more than central tendency — we need dispersion, the degree to which the data spread away from the centre. The smaller the dispersion, the more concentrated and consistent the data; the larger, the more uneven. Reporting only the mean without dispersion tells half the story.
2The most intuitive measure is the range: maximum minus minimum. It is quick, but decided by just two extreme values, and a single outlier distorts it badly. An improvement is the inter-quartile range: order the data, split into quarters, and take the spread of the middle half — the third quartile minus the first quartile — which sidesteps the extremes. A box-and-whisker diagram draws the five key values on one line, making groups easy to compare.
3Variance and standard deviation are the most widely used measures. To compute the variance, subtract the mean from each value to get the deviations, square them and average; the square root of the variance is the standard deviation. Squaring stops positive and negative deviations from cancelling, and magnifies the influence of values far from the centre. The standard deviation shares the unit of the original data; the larger it is, the more spread out the data — the more uneven the scores of a class, the larger its standard deviation.
4The standard score, also called the z-score, converts a raw mark into a relative position: subtract the mean, then divide by the standard deviation. A standard score of zero means exactly average; positive means above average, negative below. For example, scoring 80 in maths with class mean 65 and standard deviation 10 gives a standard score of 1.5; scoring 75 in English with mean 60 and standard deviation 5 gives 3. The raw mark in English is lower, yet the standard score is higher — the relative performance is stronger.
5Adding the same constant to every value leaves all dispersion unchanged; multiplying by the same positive constant scales it proportionally. For example, if everyone gains 5 marks, the range, variance and standard deviation stay the same, because the data merely shift as a whole; if every score is doubled, the standard deviation doubles too. With these two rules, score-adjustment questions become routine. Module 1 will later combine standard scores with interval proportions in the normal distribution, so the calculations of this lesson must become fluent first.
1Summary 重點摘要
The mean only captures central tendency; dispersion describes how spread out data are, with range, inter-quartile range, variance and standard deviation each playing a part — and the standard score converts raw marks into relative positions so results from different subjects can be compared fairly.
2Vocabulary cards Vocabulary
The degree to which data spread away from the centre, paired with central tendency as the two key features of a data set.
Maximum minus minimum; the quickest measure but the most sensitive to extreme values.
The third quartile minus the first quartile — the spread of the middle half of the data, robust against extremes.
A diagram showing the minimum, Q1, median, Q3 and maximum on one line, handy for comparing groups; also called a box plot.
The average of the squared deviations, measuring spread; equal to the square of the standard deviation.
The positive square root of the variance, sharing the unit of the data; the larger it is, the more spread out the data.
The relative position obtained by subtracting the mean and dividing by the standard deviation; zero means exactly average.
The sum of all values divided by their count — the common measure of central tendency.
The tendency of data to cluster around a centre, described by the mean, median and similar measures.
A collection of values obtained from one observation or survey.
3Step by step Steps
- Step 1: Why dispersion is neededLooking only at the mean hides vital information. For example, two groups of five each with mean 70: group A scores 68, 69, 70, 71, 72 — everyone near the mean; group B scores 40, 50, 70, 90, 100 — split at the extremes. For teaching, the first suits a common pace while the second needs tiered support. Clearly, describing a data set requires reporting not only the central tendency (the mean) but also the spread — the dispersion; together they give the complete picture. This is the bedrock of the statistics half of Module 1 and a recurring exam focus.
- Step 2: Range and inter-quartile rangeThe range is maximum minus minimum — the fastest calculation. For example, for 3, 7, 8, 15, 20 the range is 20 − 3 = 17. But it depends only on the two ends: if 20 becomes 200, the range balloons to 197 while every other value is unchanged — no resistance to outliers at all. The inter-quartile range orders the data and takes the middle half: IQR = Q3 − Q1. In the same data Q1 = 7 and Q3 = 15, so IQR = 8, far less affected by extremes. A box-and-whisker diagram puts the minimum, Q1, median, Q3 and maximum on one line, so two groups can be compared at a glance.
- Step 3: Computing variance and standard deviationThree steps: find the mean μ; compute each deviation x − μ; square the deviations and average them to get the variance, whose square root is the standard deviation. For example, for 2, 4, 6, 8, 10: the mean is 6, the deviations −4, −2, 0, 2, 4, their squares 16, 4, 0, 4, 16, so the variance = 40 ÷ 5 = 8 and the standard deviation = √8 ≈ 2.83. Squaring prevents positive and negative deviations from cancelling and magnifies far-from-centre values, so the standard deviation is more sensitive to outliers than the IQR — good for revealing anomalies, but also easily inflated by them.
- Step 4: Comparing two data sets with dispersionCompare groups using both mean and dispersion. For example, class A scores 60, 65, 70, 75, 80: mean 70, sum of squared deviations = 100 + 25 + 0 + 25 + 100 = 250, variance = 50, SD ≈ 7.1. Class B scores 55, 65, 70, 75, 85: mean also 70, sum of squared deviations = 225 + 25 + 0 + 25 + 225 = 500, variance = 100, SD = 10. Same mean, but class A has the smaller standard deviation and hence the more consistent results, while class B is more spread out. Write the full conclusion — 'same mean but class A has smaller SD, so more stable'; numbers without a conclusion lose marks.
- Step 5: Defining and computing the standard scoreThe standard score z = (x − μ)/σ divides 'how far from the mean' by 'how far values typically are', giving a unit-free relative position. For example, 80 marks in maths with mean 65 and SD 10 gives z = (80 − 65)/10 = 1.5 — 1.5 standard deviations above average; 75 in English with mean 60 and SD 5 gives z = (75 − 60)/5 = 3. Though the raw English mark is lower, its standard score is higher — the relative performance is better. Across subjects or classes, only standard scores compare fairly; schools use exactly this conversion to moderate internal rankings, and extended papers test it every year.
- Step 6: Effects of adding and multiplying by a constantAdding the same constant k to every value: the mean gains k, but each deviation x − μ is unchanged, so variance and SD are unchanged. Multiplying every value by a positive constant k: the mean and SD are both multiplied by k, and the variance by k². For example, if everyone gains 5 marks the SD stays the same; converting Celsius to Fahrenheit (multiply by 95, add 32) multiplies the SD by 95 and the variance by 8125. In questions, first decide whether the transformation is a shift or a scaling, then apply the rule; 'everyone loses 3 marks — what happens to the SD?' The answer is 'unchanged', though many wrongly say it shrinks — exactly the trap setters set.
- Step 7: Dispersion in the real worldDispersion and standard scores are used everywhere. For example, quality control uses the standard deviation to gauge production stability: the smaller the SD of screw diameters, the more uniform the product and the lower the defect rate; weather reports comparing daily temperature ranges between cities are applying the range; schools compare a student's relative performance across subjects with standard scores when deciding promotion or scholarships. Module 1 will later pair standard scores with the 68–95–99.7 rule in the normal distribution to estimate interval proportions; master this lesson's variance and z-score computations now and the later topic follows naturally.
5Worked example Example
Example: classes A and B, five students each, sit the same test. Class A: 60, 65, 70, 75, 80; class B: 55, 65, 70, 75, 85. (a) Find each class's mean and standard deviation. (b) Compare the two classes. (c) Siu Wah scores 80 in class A and Siu Keung scores 85 in class B; use standard scores to decide who performs relatively better.
- (a) Mean of A = (60 + 65 + 70 + 75 + 80) ÷ 5 = 70; mean of B = (55 + 65 + 70 + 75 + 85) ÷ 5 = 70
- Class A sum of squared deviations = (−10)² + (−5)² + 0² + 5² + 10² = 250, variance = 250 ÷ 5 = 50, SD = √50 ≈ 7.1
- Class B sum of squared deviations = (−15)² + (−5)² + 0² + 5² + 15² = 500, variance = 500 ÷ 5 = 100, SD = 10
- (b) The means are both 70, but class A has the smaller standard deviation, so class A is more consistent while class B is more spread out
- (c) Siu Wah's standard score = (80 − 70)/√50 ≈ 1.41; Siu Keung's = (85 − 70)/10 = 1.5; since 1.5 > 1.41, Siu Keung performs relatively better
- Answer: Both classes have mean 70; class A has SD ≈ 7.1 and class B SD = 10, so class A is more consistent. Siu Wah's standard score ≈ 1.41 and Siu Keung's = 1.5, so Siu Keung performs relatively better. In the exam, show the mean and SD calculations, then state the conclusion as 'same mean but smaller standard deviation'; for cross-class comparison use standard scores rather than raw marks.
6Common mistakes Mistakes
7Quick check Quick check
Q1What is dispersion, and why is the mean alone not enough to describe a data set?
Q2Find the mean and variance of 4, 8, 10, 14, 14.
Q3Find the standard deviation of 2, 4, 6, 8, 10.
Q4A subject has mean 62 and SD 8; Siu Ting scores 78. Find her standard score.
Q5If every value of a data set is multiplied by 3, what happens to the mean, the variance and the standard deviation?
8Recap Recap
- Dispersion describes the spread of data, paired with central tendency as the two key features
- Range = maximum − minimum; inter-quartile range = Q3 − Q1, robust against extreme values
- Variance = the average squared deviation; standard deviation = its positive square root, sharing the data's unit
- Standard score z = (x − μ)/σ compares relative performance fairly across groups and subjects
- Adding a constant leaves dispersion unchanged; multiplying by a positive k multiplies the SD by k and the variance by k²