Standard Deviation Practice Questions

Standard Deviation is a measure of dispersion that indicates how much the values in a dataset typically differ from the mean.

Example 1: Calculate the population standard deviation for the following data set: {6, 8, 10, 12, 14}

Solution:

Mean = (6 + 8 + 10 + 12 + 14) / 5 = 50 / 5 = 10

Now, calculate the squared deviations from the mean,

• (6 - 10)² = 16
• (8 - 10)² = 4
• (10 - 10)² = 0
• (12 - 10)² = 4
• (14 - 10)² = 16

Thus, Σ(xi - μ)² = 16 + 4 + 0 + 4 + 16 = 40

Divide the sum of squared differences by N - 1, and take square root to find standard deviation,

σ = √(40/5) = √8 ≈ 2.83

So, the population standard deviation of the data set is approximately 2.83.

Example 2: Calculate the sample standard deviation for the following data set: {12, 15, 18, 21, 24}

Solution:

Mean = (12 + 15 + 18 + 21 + 24) / 5 = 90 / 5 = 18

Now, calculate the squared deviations from the mean,

• (12 - 18)² = 36
• (15 - 18)² = 9
• (18 - 18)² = 0
• (21 - 18)² = 9
• (24 - 18)² = 36

Thus, Σ(xi - μ)² = 36 + 9 + 0 + 9 + 36 = 90

Divide the sum of squared differences by N - 1, and take square root to find standard deviation,

s = √(90/4) = √22.5 ≈ 4.74

Therefore, the sample standard deviation of the data set is approximately 4.74.

Example 3: Find the Standard Deviation of the following data

Solution:

First, make the table as follows, so we can calculate the further values easily.

Xi	fi	Xi×fi	Xi-μ	(Xi-μ)2	f×(Xi-μ)2
5	2	10	-6.375	40.64	81.28
12	3	36	0.625	0.39	1.17
15	3	45	3.625	13.14	39.42
Total	8	91			121.87

Mean (μ) = ∑(f_i x_i)/∑(f_i)

⇒ Mean (μ) = 91/8 = 11.375

using standard deviation formula

σ = √(∑_iⁿ f_i(x_i - μ)²/n)
⇒ σ = √[(121.87)/(8)]
⇒ σ = √(15.234)
⇒ σ  = 3.90

Standard Derivation(σ) = 3.90

Example 4: Find the Standard Deviation of the following data table.

Solution:

Class	Xi	fi	f×Xi	Xi - μ	(Xi - μ)2	f×(Xi - μ)2
0-10	5	3	15	-15	225	675
10-20	15	6	90	-5	25	150
20-30	25	4	100	5	25	100
30-40	35	2	70	15	225	450
40-50	45	1	45	25	625	625
Total		16	320			2000

Mean (μ) = ∑(fi xi)/∑(fi)

⇒ Mean (μ) = 320/16 = 20

now, by using standard deviation formula

σ = √(∑_iⁿ f_i(x_i - μ)²/n)
⇒ σ = √[(2000)/(16)]
⇒ σ = √(125)
⇒ σ = 11.18

Standard Derivation(σ) = 11.18

Practice Problems

Problem 1: Given the test scores of a class, 65, 70, 78, 72, 68, 74, 81, 70, calculate the standard deviation.

Problem 2: A farmer measures the weight of ten pumpkins in pounds: 12, 15, 17, 11, 16, 14, 15, 16, 14, 15. Compute the standard deviation to understand the variability in pumpkin weights.

Problem 3: Two teachers recorded the scores of their students on the same exam. Teacher A's student scores: 88, 92, 76, 94, 85. Teacher B's student scores: 85, 83, 84, 87, 86. Calculate and compare the standard deviation of scores from both classes.

Problem 4: Consider the ages of participants in a study: 34, 37, 29, 31, 38, 36, 30, 33. Calculate the standard deviation and discuss what this might suggest about the spread of ages in the study.

Problem 5: A basketball player's points per game over ten games are recorded as follows: 22, 28, 26, 32, 24, 19, 35, 27, 23, 31. Find the standard deviation to evaluate the consistency of the player's scoring.