EN
Book a Trial Class

Measures of Dispersion for Ungrouped Data · Form 4

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

The three basic tools of dispersion, range, variance and standard deviation, each shown in one clean technique. Best for students meeting the formula variance = Σx²/N − mean² for the first time.

Written by the spmmath.com.my editorial team.

Worked example 1

A trainee did push-ups on 5 days: 12, 18, 9, 15, 21. Find the range of the number of push-ups.

  1. Identify the largest value: 21, and the smallest value: 9.
  2. Range = largest value − smallest value.
  3. Range = 21 − 9 = 12.

Worked example 2

Find the variance of the data: 2, 4, 6, 8, 10.

  1. Number of data N = 5. Mean = (2+4+6+8+10)/5 = 30/5 = 6.
  2. Sum of squares Σx² = 2²+4²+6²+8²+10² = 4+16+36+64+100 = 220.
  3. Apply variance = Σx²/N − mean² = 220/5 − 6².
  4. = 44 − 36 = 8.

Worked example 3

Find the standard deviation of the data: 2, 4, 5, 6, 8.

  1. N = 5. Mean = (2+4+5+6+8)/5 = 25/5 = 5.
  2. Σx² = 2²+4²+5²+6²+8² = 4+16+25+36+64 = 145.
  3. Variance = Σx²/N − mean² = 145/5 − 5² = 29 − 25 = 4.
  4. Standard deviation = √variance = √4 = 2.

Worked example 4

The masses, in kg, of six school bags are 7, 3, 11, 5, 9 and 4. Find the range of the masses.

  1. Range = largest value − smallest value.
  2. Largest = 11 kg, smallest = 3 kg.
  3. Range = 11 − 3 = 8 kg.

Worked example 5

Find the variance of the data 3, 5, 7, 9.

  1. Mean = (3 + 5 + 7 + 9) ÷ 4 = 24 ÷ 4 = 6.
  2. Squared deviations: (3−6)² = 9, (5−6)² = 1, (7−6)² = 1, (9−6)² = 9.
  3. Sum of squared deviations = 9 + 1 + 1 + 9 = 20.
  4. Variance = 20 ÷ 4 = 5.

Worked example 6

Find the standard deviation of the data 1, 3, 5, 7, 9. Give your answer correct to two decimal places.

  1. Mean = (1 + 3 + 5 + 7 + 9) ÷ 5 = 25 ÷ 5 = 5.
  2. Squared deviations: 16, 4, 0, 4, 16; sum = 40.
  3. Variance = 40 ÷ 5 = 8.
  4. Standard deviation = √8 ≈ 2.83.

Worked example 7

The temperatures, in °C, recorded in a science laboratory on 5 days are 24, 27, 22, 29, 25. Find the range of the temperatures.

  1. Range = highest value − lowest value
  2. Highest = 29, Lowest = 22
  3. Range = 29 − 22 = 7°C

Worked example 8

Find the standard deviation of the data: 3, 6, 9, 12, 15. Give your answer correct to two decimal places.

  1. Mean = (3+6+9+12+15)/5 = 45/5 = 9
  2. Σx² = 3²+6²+9²+12²+15² = 9+36+81+144+225 = 495
  3. Variance = Σx²/n − mean² = 495/5 − 9² = 99 − 81 = 18
  4. Standard deviation = √18 = 4.24 (2 d.p.)

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

Book a Trial ClassOne-hour paid trial · Same-day reply · from RM50/hr

Frequently asked questions

What should I focus on for easy questions on measures of dispersion (ungrouped data)?

Easy questions ask you to find the range, variance, or standard deviation directly from a small list of ungrouped data or a simple frequency table. Learn the standard deviation formula properly and use your calculator's statistics mode carefully, a wrong mode setting is the most common reason a correct method gives a wrong final answer.

Why do range and standard deviation sometimes give a different sense of "spread" for the same data?

Range only compares the highest and lowest values, so one extreme outlier can make the range look large even if most data points are close together. Standard deviation considers every value's distance from the mean, giving a more complete picture of spread, this is why exam questions often ask you to compare both measures, not just one.

What's a common calculation mistake when finding variance by hand?

Students often forget to square each deviation from the mean before summing, or they square the mean instead of squaring each individual value correctly in the Σx² formula. Write out the formula symbolically first, fill in one value at a time in a table, and only substitute into the final formula once every column is complete.

Book a Trial Class

Book a Trial Class
One-hour paid trial · Same-day replyfrom RM50/hr
Book a Trial Class