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Measures of Dispersion for Ungrouped Data · Form 4

Measures of Dispersion for Ungrouped Data: Worked Examples (Medium)

Two- and three-step problems: find the mean first, then variance and standard deviation, and compare the spread of two data sets. Helps students link the formula to judging consistency.

Written by the spmmath.com.my editorial team.

Worked example 1

The masses (kg) of 5 melons are 4, 6, 7, 8, 10. Find the mean, then the standard deviation of the masses.

  1. N = 5. Mean = (4+6+7+8+10)/5 = 35/5 = 7 kg.
  2. Σx² = 4²+6²+7²+8²+10² = 16+36+49+64+100 = 265.
  3. Variance = Σx²/N − mean² = 265/5 − 7² = 53 − 49 = 4.
  4. Standard deviation = √4 = 2 kg.

Worked example 2

Two players record their goals over 5 matches. Player A: 3, 5, 6, 7, 9.

Player B: 2, 4, 6, 8, 10. Using the standard deviation, decide who is more consistent.

  1. Means: A = 30/5 = 6; B = 30/5 = 6, the means are equal, so compare the spread.
  2. Player A: Σx² = 9+25+36+49+81 = 200; variance = 200/5 − 6² = 40 − 36 = 4; sd = √4 = 2.
  3. Player B: Σx² = 4+16+36+64+100 = 220; variance = 220/5 − 6² = 44 − 36 = 8; sd = √8 ≈ 2.83.
  4. Player A has the smaller standard deviation, so A's goals vary less.

Worked example 3

A coach records that for 6 netball players, Σx = 60 and Σx² = 654, where x is the number of goals scored. Find the mean, variance and standard deviation.

  1. N = 6. Mean = Σx/N = 60/6 = 10.
  2. Variance = Σx²/N − mean² = 654/6 − 10² = 109 − 100 = 9.
  3. Standard deviation = √variance = √9 = 3.

Worked example 4

The number of units sold by a shop over 5 days are 8, 11, 7, 10 and 14. Using Σx and Σx², find the mean and the variance.

  1. Σx = 8 + 11 + 7 + 10 + 14 = 50, so mean = 50 ÷ 5 = 10.
  2. Σx² = 64 + 121 + 49 + 100 + 196 = 530.
  3. Variance = Σx²/n − (mean)² = 530 ÷ 5 − 10².
  4. Variance = 106 − 100 = 6.

Worked example 5

The mean of 8 numbers is 5 and the sum of the squares of the numbers, Σx², is 250. Find the standard deviation.

  1. Variance = Σx²/n − (mean)² = 250 ÷ 8 − 5².
  2. Variance = 31.25 − 25 = 6.25.
  3. Standard deviation = √6.25 = 2.5.

Worked example 6

In an archery test, Archer P scored 6, 7, 8, 9, 10 and Archer Q scored 4, 6, 8, 10, 12 over five rounds. Both have the same mean score.

By calculating the standard deviation of each, determine which archer is more consistent.

  1. Both means = 40 ÷ 5 = 8.
  2. Archer P squared deviations: 4, 1, 0, 1, 4; sum = 10; variance = 2; SD = √2 ≈ 1.41.
  3. Archer Q squared deviations: 16, 4, 0, 4, 16; sum = 40; variance = 8; SD = √8 ≈ 2.83.
  4. P has the smaller standard deviation, so Archer P is more consistent.

Worked example 7

The number of customer complaints received by a shop over 5 months has a mean of 12 and a standard deviation of 3. Find Σx and Σx² for this data.

  1. Σx = mean × n = 12 × 5 = 60
  2. Variance = (standard deviation)² = 3² = 9
  3. Σx² = (variance + mean²) × n = (9 + 12²) × 5 = (9 + 144) × 5 = 765

Worked example 8

The number of minutes 6 students spent on revision one evening were recorded as 10, 12, 14, 16, 18 and 20. Find the mean revision time, then the variance of the data, correct to two decimal places.

  1. Mean = (10+12+14+16+18+20)/6 = 90/6 = 15 minutes
  2. Σx² = 10²+12²+14²+16²+18²+20² = 100+144+196+256+324+400 = 1420
  3. Variance = Σx²/n − mean² = 1420/6 − 15² = 236.67 − 225 = 11.67 (2 d.p.)

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

What formulas do I need for finding variance and standard deviation of ungrouped data?

You need variance = Σ(x − x̄)²/n (or the shortcut Σx²/n − x̄²) and standard deviation = √variance, where x̄ is the mean and n is the number of data. Learn both versions of the variance formula, the shortcut is faster once you already have Σx and Σx².

How do I compare the consistency of two sets of data using standard deviation?

Calculate the mean and standard deviation for each set. The set with the smaller standard deviation is more consistent, because its values are closer to its own mean.

Always state both the mean (which set scores higher) and the standard deviation (which set is more consistent), a full comparison needs both.

What mistakes do students usually make in medium-level dispersion questions?

The most common errors are forgetting to square-root the variance to get standard deviation, substituting Σx² incorrectly, and rounding too early in a multi-step calculation. Some students also mix up variance and standard deviation when writing their final answer, check which one the question actually asks for.

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