Standard Deviation Questions and Solutions

Grasping the concept of Standard Deviation is made easier with the help of well-crafted questions and their detailed solutions. The standard deviation is a statistical measure that shows the dispersion of a dataset relative to its mean. The questions and solutions provided here will offer a comprehensive understanding of the concept and will aid in practicing and mastering it. For each question, complete explanations are provided to help you cross-check your answers. For more insights on standard deviation, click here .

What is Standard Deviation?

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Standard deviation is a statistical measure used in descriptive statistics to understand the dispersion of data points in relation to the mean. It calculates the deviation of data points from the mean and shows how these values are distributed in a data sample. The standard deviation of a data set is the square root of its variance.

How to Calculate Standard Deviation?

Standard Deviation – Practice Questions

1. Find the standard deviation of the data set:
   4, 8, 6, 5, 3, 7, 9
    
2. Calculate the standard deviation of the numbers:
   15, 21, 18, 24, 30
    
3. Find the standard deviation for the following values:
   10, 12, 14, 16, 18
    
4. A student scored the following marks in five tests:
   78, 85, 90, 83, 88
   What is the standard deviation of the marks?
    
5. The heights (in cm) of 6 students are:
   150, 152, 155, 158, 160, 162

You can also explore Variance and Standard Deviation for more information.

1. Why is Standard Deviation the Most Reliable Measure of Spread?

Answer:
Standard deviation is the most accurate way to measure how spread out data is because:

• It considers every data point, giving a true sense of variability.
   

• A small change in one value affects the result, making it sensitive and detailed.
   

• It's useful for comparing two different data groups.
   

• It's essential in advanced statistical tests like hypothesis testing.

2. What Does a Standard Deviation of Zero Mean?

Answer:
If a group of students has a mean score of 75 and the standard deviation is 0, it means everyone scored exactly 75. No one scored more or less, so there’s no spread in the data.

3. Find the Standard Deviation of These Heights (in cm): 42, 55, 49, 63, 58

Solution:

• Total plants = 5

• Mean = (42 + 55 + 49 + 63 + 58) / 5 = 267 / 5 = 53.4

• Use the formula:
  SD = sqrt [ ( (x1 − x̄)² + (x2 − x̄)² + (x3 − x̄)² + ... + (xn − x̄)² ) / (n − 1) ]

• Differences squared:
  (42–53.4)² + (55–53.4)² + (49–53.4)² + (63–53.4)² + (58–53.4)² = 129.96 + 2.56 + 19.36 + 92.16 + 21.16 = 265.2

• SD = √(265.2 / 4) = √66.3 ≈ 8.14

4. Prove the SD of the First n Natural Numbers = √[(n²–1)/12]

Solution:

• Mean = (n + 1)/2
• ∑x² = n(n+1)(2n+1)/6
• Use formula:
  SD = √[ (∑x² / n) − ( (∑x / n) )² ]

We know:

• ∑x = n(n + 1)/2
• ∑x² = n(n + 1)(2n + 1)/6

Substitute these into the formula:

SD = √[ {n(n + 1)(2n + 1)/6n} − { (n + 1)² / 4 } ]

Simplify the expression:

SD = √[ (n² − 1) / 12 ]

5. For X: 1, 2, 3 with P(X): 0.2, 0.5, 0.3, Find the SD

Solution:

• Mean = 1×0.2 + 2×0.5 + 3×0.3 = 2.1
• E(X²) = 1²×0.2 + 2²×0.5 + 3²×0.3 = 0.2 + 2 + 2.7 = 4.9
• Var(X) = 4.9 – (2.1)² = 4.9 – 4.41 = 0.49
• SD = √0.49 = 0.7

6. Rolling Two Dice: Find the Variance and SD of the Total Outcome

Solution:

• Possible sums range from 2 to 12.
• Probabilities:
  • P(2) = 1/36, P(3) = 2/36, ..., P(12) = 1/36
• Mean = 7 (known result)
• E(X²) = Approx. 54.83
• Variance = 54.83 – 7² = 54.83 – 49 = 5.83
• SD = √5.83 ≈ 2.41

7. X = 2, 4, 6 with P(X) = 0.4, 0.4, 0.2. Find the SD

Solution:

• Mean = 2×0.4 + 4×0.4 + 6×0.2 = 3.6
• E(X²) = 4×0.4 + 16×0.4 + 36×0.2 = 1.6 + 6.4 + 7.2 = 15.2
• Var(X) = 15.2 – (3.6)² = 15.2 – 12.96 = 2.24
• SD = √2.24 ≈ 1.5

8. Find SD for: 5, 7, 10, 12 Using the Actual Mean Method

Solution:

• Mean = (5 + 7 + 10 + 12) / 4 = 8.5
• Squared differences = (5–8.5)² + (7–8.5)² + (10–8.5)² + (12–8.5)²
  = 12.25 + 2.25 + 2.25 + 12.25 = 29
• Variance = 29 / 4 = 7.25
• SD = √7.25 ≈ 2.69

9. SD of First 6 Natural Numbers

Solution:

SD = √[ (n² − 1) / 12 ]

Step 1: Substitute n = 6 into the formula:
SD = √[ (6² − 1) / 12 ]

Step 2: Calculate the expression inside the square root:
SD = √[ (36 − 1) / 12 ]
SD = √[ 35 / 12 ]

Step 3: Simplify the result:
SD = √(2.9167)
SD ≈ 1.71

10. Find SD of 15, 18, 12, 22, 24

Solution:

• Mean = (15 + 18 + 12 + 22 + 24)/5 = 91/5 = 18.2
• Squared differences = (15–18.2)² + (18–18.2)² + (12–18.2)² + (22–18.2)² + (24–18.2)²
  = 10.24 + 0.04 + 38.44 + 14.44 + 33.64 = 96.8
• Variance = 96.8 / 5 = 19.36
• SD = √19.36 ≈ 4.4

Questions for Practice

1. A student mistakenly recorded one observation as 50 instead of 40. The mean and standard deviation calculated for 100 observations were 40 and 5.1, respectively. Determine the correct mean and standard deviation.
2. In a class of 50 students, four students were randomly selected and their final evaluation scores were recorded as 812, 982, 836, and 769. Calculate the standard deviation of their scores.
3. The average daily wage of 50 factory workers is Rs. 200 with a standard deviation of Rs. 40. Each worker is offered a raise of Rs. 20. What is the new daily average wage and standard deviation?

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