Solving Statistics Problems On Variance

 

Understanding Variance

Variance is a statistical measurement of the spread between numbers in a data set. More specifically, it measures how far each number in the set is from the mean (average), and therefore from every other number in the set. A variance of 0 indicates that all values are identical.

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Formulas for Variance

1. Ungrouped Data (Sample Variance)

For a set of individual raw scores, the formula is:

s² = Σ(x - x̄)² / (n - 1)

Where:
• s² = Sample variance
• Σ = Summation (add them all up)
• x = Each individual value
• x̄ = Mean of the values
• n = Total number of values

2. Grouped Data (Sample Variance)

When data is organized into frequency distributions, the formula is:

s² = Σf(x - x̄)² / (Σf - 1)

Where:
• s² = Sample variance
• f = Frequency of each class/value
• x = Midpoint of the class interval (or the specific value)
• x̄ = Grouped mean (Σfx / Σf)
• Σf = Total frequency (n)

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Practice Quiz: Variance for Ungrouped Data

Test your knowledge with these 5 objective questions. Click the button under each question to reveal the correct answer and step-by-step solution.

Q1. Find the variance of the following data set: 2, 4, 6, 8, 10. (Assume this is a sample).

• A) 4
• B) 8
• C) 10
• D) 40

Correct Answer: B) 8

Step-by-step Solution:
1. Find the Mean (x̄): (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6.
2. Subtract the Mean from each value and square the result:
• (2 - 6)² = 16
• (4 - 6)² = 4
• (6 - 6)² = 0
• (8 - 6)² = 4
• (10 - 6)² = 16
3. Sum the squared differences: 16 + 4 + 0 + 4 + 16 = 40.
4. Divide by (n - 1) for sample variance: 40 / (5 - 1) = 40 / 4 = 8.

Q2. If a data set has a variance of 0, what does this imply about the data points?

• A) All data points are equal to zero.
• B) Half of the data points are negative.
• C) All data points are identical to each other.
• D) The mean of the data set is 0.

Correct Answer: C) All data points are identical to each other.

Step-by-step Solution:
Variance measures the spread of data points from the mean. If there is no spread at all (variance = 0), it means every single data point is exactly equal to the mean, making them all identical to each other.

Q3. Calculate the sample variance for the data points: 5, 5, 5, 5.

• A) 0
• B) 5
• C) 20
• D) 1.25

Correct Answer: A) 0

Step-by-step Solution:
1. Find the Mean (x̄): (5 + 5 + 5 + 5) / 4 = 20 / 4 = 5.
2. Because every number is equal to the mean, the difference (x - x̄) for each number is 0.
3. The sum of squared deviations is 0, so the variance is 0.

Q4. For the sample data set 3, 6, 9, what is the value of the numerator in the variance formula, Σ(x - x̄)²?

• A) 6
• B) 9
• C) 18
• D) 27

Correct Answer: C) 18

Step-by-step Solution:
1. Find the Mean (x̄): (3 + 6 + 9) / 3 = 18 / 3 = 6.
2. Compute (x - x̄)² for each:
• (3 - 6)² = (-3)² = 9
• (6 - 6)² = (0)² = 0
• (9 - 6)² = (3)² = 9
3. Sum them up: 9 + 0 + 9 = 18.

Q5. If every value in a sample data set is multiplied by 2, what happens to the variance of the data set?

• A) The variance remains the same.
• B) The variance is multiplied by 2.
• C) The variance is multiplied by 4.
• D) The variance is divided by 2.

Correct Answer: C) The variance is multiplied by 4.

Step-by-step Solution:
Variance is a squared property (s²). When you scale data by a constant factor k, the new variance becomes k² times the original variance. Since the factor is 2, the variance changes by 2² = 4.