Medium Variance Calculation Practice Questions

Understanding how to measure the spread of data is a fundamental skill in statistics. A key metric for this is variance, which quantifies the dispersion of data points around the mean. Mastering variance calculation is essential for further topics in statistical analysis, such as hypothesis testing and regression. This guide provides a clear breakdown of the concept, worked examples, and a range of practice problems to build your confidence.

Concept Explanation

Variance is a statistical measurement of the spread between numbers in a data set from their average value. In simpler terms, it calculates how far each number in the set is from the mean (average) and thus from every other number in the set. A low variance indicates that the data points tend to be very close to the mean and hence to each other, while a high variance indicates that the data points are spread out over a wider range of values. The process of finding this value is known as variance calculation.

There are two primary types of variance calculation, depending on whether you are working with an entire population or a sample of that population:

1. Population Variance (σ²)

You use this formula when your dataset includes every member of the group you are interested in.

The formula for population variance is:

σ² = Σ (xᵢ - μ)² / N

• σ² is the population variance.
• Σ is the summation symbol, meaning "sum up."
• xᵢ represents each individual data point.
• μ (mu) is the population mean.
• N is the total number of data points in the population.

2. Sample Variance (s²)

You use this formula when you have a smaller group (a sample) taken from a larger population. This is more common in real-world data analysis.

The formula for sample variance is:

s² = Σ (xᵢ - x̄)² / (n - 1)

• s² is the sample variance.
• Σ is the summation symbol.
• xᵢ represents each individual data point in the sample.
• x̄ (x-bar) is the sample mean.
• n is the number of data points in the sample.

The key difference is the denominator: (n-1). This is known as Bessel's correction, which is used to provide a more accurate and unbiased estimate of the true population variance when working with a sample. Understanding how to calculate the mean is a prerequisite for variance calculation, and you can practice it on our Mean, Median, Mode Practice Questions page.

Solved Examples of Variance Calculation

The best way to understand how to perform a variance calculation is to walk through some examples. Here are a few solved problems demonstrating the process for both population and sample data.

Example 1: Population Variance

Problem: A small company has 5 employees. Their annual salaries (in thousands of dollars) are 50, 55, 60, 65, and 70. Calculate the population variance of their salaries.

Solution:

1. Calculate the population mean (μ): The mean is the sum of the values divided by the number of values (N).

   μ = (50 + 55 + 60 + 65 + 70) / 5 = 300 / 5 = 60

2. Calculate the squared differences from the mean for each data point: Subtract the mean from each value and square the result.
   • (50 - 60)² = (-10)² = 100
   • (55 - 60)² = (-5)² = 25
   • (60 - 60)² = (0)² = 0
   • (65 - 60)² = (5)² = 25
   • (70 - 60)² = (10)² = 100
3. Sum the squared differences:

   Σ (xᵢ - μ)² = 100 + 25 + 0 + 25 + 100 = 250

4. Divide by the total number of data points (N):

   σ² = 250 / 5 = 50

The population variance of the salaries is 50 (in thousands of dollars squared).

Example 2: Sample Variance

Problem: A researcher measures the height of 6 randomly selected students from a large university. Their heights in inches are 64, 66, 68, 70, 72, and 74. Calculate the sample variance of their heights.

Solution:

1. Calculate the sample mean (x̄):

   x̄ = (64 + 66 + 68 + 70 + 72 + 74) / 6 = 414 / 6 = 69

2. Calculate the squared differences from the mean:
   • (64 - 69)² = (-5)² = 25
   • (66 - 69)² = (-3)² = 9
   • (68 - 69)² = (-1)² = 1
   • (70 - 69)² = (1)² = 1
   • (72 - 69)² = (3)² = 9
   • (74 - 69)² = (5)² = 25
3. Sum the squared differences:

   Σ (xᵢ - x̄)² = 25 + 9 + 1 + 1 + 9 + 25 = 70

4. Divide by (n - 1): Since this is a sample, we divide by the number of data points minus one (6 - 1 = 5).

   s² = 70 / 5 = 14

The sample variance of the students' heights is 14 inches squared.

Example 3: Variance with a Non-Integer Mean

Problem: Find the sample variance for the following set of quiz scores: 15, 17, 12, 18, 11.

Solution:

1. Calculate the sample mean (x̄):

   x̄ = (15 + 17 + 12 + 18 + 11) / 5 = 73 / 5 = 14.6

2. Calculate the squared differences from the mean:
   • (15 - 14.6)² = (0.4)² = 0.16
   • (17 - 14.6)² = (2.4)² = 5.76
   • (12 - 14.6)² = (-2.6)² = 6.76
   • (18 - 14.6)² = (3.4)² = 11.56
   • (11 - 14.6)² = (-3.6)² = 12.96
3. Sum the squared differences:

   Σ (xᵢ - x̄)² = 0.16 + 5.76 + 6.76 + 11.56 + 12.96 = 37.2

4. Divide by (n - 1): We have 5 data points, so we divide by (5 - 1 = 4).

   s² = 37.2 / 4 = 9.3

The sample variance of the quiz scores is 9.3.

Practice Questions

Now it's your turn to practice. Use the appropriate formulas and remember to distinguish between population and sample data.

1. (Easy) An entire class of 4 students took a test. Their scores were 80, 85, 90, and 95. What is the population variance of their scores?

2. (Easy) A botanist measures the petal length (in cm) of a random sample of 5 flowers from a large field. The lengths are 4, 5, 6, 7, and 8. What is the sample variance?

3. (Medium) Calculate the population variance for the dataset: {2, 3, 6, 8, 11}.

4. (Medium) A quality control inspector takes a sample of 7 light bulbs and records their lifespan in hours: 1010, 990, 1020, 980, 1000, 1030, 970. What is the sample variance of the lifespans?

5. (Medium) The daily high temperatures (°C) for a city over an entire week were recorded as: 22, 25, 19, 20, 23, 26, 21. Treating this as a population for that specific week, what is the variance?

6. (Medium) A sample of 6 laptops has the following battery lives in hours: 8.5, 9.0, 7.5, 8.0, 9.5, 8.5. Calculate the sample variance.

7. (Hard) Two basketball players, Alex and Ben, have their points per game from a sample of 5 games recorded. Alex's scores: {15, 25, 20, 18, 22}. Ben's scores: {20, 21, 19, 20, 20}. Calculate the sample variance for each player. Which player is more consistent based on the variance?

8. (Hard) A dataset has a sample mean (x̄) of 50 and a sample size (n) of 10. The sum of the squared differences from the mean, Σ(xᵢ - x̄)², is 720. What is the sample variance?

9. (Hard) You are given the following population data: {1, 1, 1, 5, 5, 5}. What is the population variance?

10. (Hard) A sample of data has a variance of 15. If every value in the dataset is increased by 10, what will the new sample variance be?

Answers & Explanations

1. Answer: 31.25

Explanation: This is a population (N=4).

1. Mean (μ): (80 + 85 + 90 + 95) / 4 = 350 / 4 = 87.5
2. Squared Differences: (80 - 87.5)² = (-7.5)² = 56.25 (85 - 87.5)² = (-2.5)² = 6.25 (90 - 87.5)² = (2.5)² = 6.25 (95 - 87.5)² = (7.5)² = 56.25
3. Sum of Squares: 56.25 + 6.25 + 6.25 + 56.25 = 125
4. Population Variance (σ²): 125 / 4 = 31.25

2. Answer: 2.5

Explanation: This is a sample (n=5).

1. Mean (x̄): (4 + 5 + 6 + 7 + 8) / 5 = 30 / 5 = 6
2. Squared Differences: (4 - 6)² = (-2)² = 4 (5 - 6)² = (-1)² = 1 (6 - 6)² = 0² = 0 (7 - 6)² = 1² = 1 (8 - 6)² = 2² = 4
3. Sum of Squares: 4 + 1 + 0 + 1 + 4 = 10
4. Sample Variance (s²): 10 / (5 - 1) = 10 / 4 = 2.5

3. Answer: 10.8

Explanation: This is a population (N=5).

1. Mean (μ): (2 + 3 + 6 + 8 + 11) / 5 = 30 / 5 = 6
2. Squared Differences: (2 - 6)² = (-4)² = 16 (3 - 6)² = (-3)² = 9 (6 - 6)² = 0² = 0 (8 - 6)² = 2² = 4 (11 - 6)² = 5² = 25
3. Sum of Squares: 16 + 9 + 0 + 4 + 25 = 54
4. Population Variance (σ²): 54 / 5 = 10.8

4. Answer: 466.67

Explanation: This is a sample (n=7).

1. Mean (x̄): (1010 + 990 + 1020 + 980 + 1000 + 1030 + 970) / 7 = 7000 / 7 = 1000
2. Squared Differences: (1010 - 1000)² = 10² = 100 (990 - 1000)² = (-10)² = 100 (1020 - 1000)² = 20² = 400 (980 - 1000)² = (-20)² = 400 (1000 - 1000)² = 0² = 0 (1030 - 1000)² = 30² = 900 (970 - 1000)² = (-30)² = 900
3. Sum of Squares: 100 + 100 + 400 + 400 + 0 + 900 + 900 = 2800
4. Sample Variance (s²): 2800 / (7 - 1) = 2800 / 6 ≈ 466.67

5. Answer: 6.29

Explanation: This is a population for the week (N=7).

1. Mean (μ): (22 + 25 + 19 + 20 + 23 + 26 + 21) / 7 = 156 / 7 ≈ 22.29
2. Squared Differences (using ≈ 22.29 for mean): (22 - 22.29)² ≈ 0.0841 (25 - 22.29)² ≈ 7.3441 (19 - 22.29)² ≈ 10.8241 (20 - 22.29)² ≈ 5.2441 (23 - 22.29)² ≈ 0.5041 (26 - 22.29)² ≈ 13.7641 (21 - 22.29)² ≈ 1.6641
3. Sum of Squares: 0.0841 + 7.3441 + 10.8241 + 5.2441 + 0.5041 + 13.7641 + 1.6641 = 44 (approximately, using fractions gives an exact sum)
4. Population Variance (σ²): 44 / 7 ≈ 6.29

6. Answer: 0.5

Explanation: This is a sample (n=6).

1. Mean (x̄): (8.5 + 9.0 + 7.5 + 8.0 + 9.5 + 8.5) / 6 = 51 / 6 = 8.5
2. Squared Differences: (8.5 - 8.5)² = 0² = 0 (9.0 - 8.5)² = 0.5² = 0.25 (7.5 - 8.5)² = (-1.0)² = 1.0 (8.0 - 8.5)² = (-0.5)² = 0.25 (9.5 - 8.5)² = 1.0² = 1.0 (8.5 - 8.5)² = 0² = 0
3. Sum of Squares: 0 + 0.25 + 1.0 + 0.25 + 1.0 + 0 = 2.5
4. Sample Variance (s²): 2.5 / (6 - 1) = 2.5 / 5 = 0.5

7. Answer: Alex's variance is 17.5. Ben's variance is 0.5. Ben is more consistent.

Explanation: Both are samples (n=5). A lower variance indicates more consistency.

For Alex:

1. Mean (x̄): (15 + 25 + 20 + 18 + 22) / 5 = 100 / 5 = 20
2. Sum of Squares: (15-20)² + (25-20)² + (20-20)² + (18-20)² + (22-20)² = 25 + 25 + 0 + 4 + 4 = 58
3. Sample Variance (s²): 58 / (5 - 1) = 58 / 4 = 14.5

For Ben:

1. Mean (x̄): (20 + 21 + 19 + 20 + 20) / 5 = 100 / 5 = 20
2. Sum of Squares: (20-20)² + (21-20)² + (19-20)² + (20-20)² + (20-20)² = 0 + 1 + 1 + 0 + 0 = 2
3. Sample Variance (s²): 2 / (5 - 1) = 2 / 4 = 0.5

Ben's scores have a much lower variance (0.5 vs. 14.5), so his performance is significantly more consistent.

8. Answer: 80

Explanation: This question tests your understanding of the formula itself. You are given the key components.

Sample Variance Formula: s² = Σ (xᵢ - x̄)² / (n - 1)

You are given: Σ (xᵢ - x̄)² = 720 and n = 10.

s² = 720 / (10 - 1) = 720 / 9 = 80

9. Answer: 4

Explanation: This is a population (N=6).

1. Mean (μ): (1 + 1 + 1 + 5 + 5 + 5) / 6 = 18 / 6 = 3
2. Squared Differences: (1 - 3)² = (-2)² = 4 (This occurs 3 times) (5 - 3)² = (2)² = 4 (This occurs 3 times)
3. Sum of Squares: (3 * 4) + (3 * 4) = 12 + 12 = 24
4. Population Variance (σ²): 24 / 6 = 4

10. Answer: 15

Explanation: Variance is a measure of spread or dispersion. Adding a constant value to every data point shifts the entire dataset, including the mean, by that constant. However, the distance between each point and the mean remains exactly the same. Since variance is calculated from these distances (the spread), it does not change. Therefore, the new variance is still 15. Once you've mastered variance, you can learn about standard deviation, which is simply the square root of variance and often easier to interpret.

Quick Quiz

Frequently Asked Questions

What is the difference between population and sample variance?

Population variance (σ²) measures the spread of data for an entire group of interest, using N (the total number of items) in the denominator. Sample variance (s²) measures the spread of data for a subset (sample) of a larger population and uses (n-1) in the denominator to provide a better estimate of the true population variance.

Why do we divide by n-1 for sample variance?

We divide by n-1, a technique called Bessel's correction, because a sample's variance tends to underestimate the true population's variance. Using n-1 instead of n in the denominator inflates the result slightly, making it a more accurate, or unbiased, estimator of the population variance. For more details on this topic, educational resources like Khan Academy offer excellent video explanations.

What does a high variance indicate?

A high variance indicates that the data points in a set are spread out far from the mean and from each other. This signifies high variability or volatility in the data. For example, in finance, a stock with a high variance in its price is considered more risky than one with a low variance.

Can variance be negative?

No, variance can never be negative. The calculation involves squaring the differences between each data point and the mean. Since the square of any real number (positive or negative) is always non-negative, the sum of these squared differences will also be non-negative. The only time variance is zero is when all data points are identical.

How is variance related to standard deviation?

Standard deviation is the square root of the variance. While variance gives a measure of spread in squared units, the standard deviation converts this back into the original units of the data, making it more intuitive to interpret. Both are fundamental to more advanced topics like finding z-scores and performing hypothesis tests.

What are the units of variance?

The units of variance are the square of the original data's units. For example, if you are measuring heights in centimeters (cm), the variance will be in square centimeters (cm²). This is one reason why standard deviation (which has the same units as the original data) is often preferred for interpretation.

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