Variance Calculation Quiz: Calculating Variance (Basic Level)

Option A describes the mean, not the variance. Option B sums raw deviations, which always total zero and produce no useful result. Option C uses absolute values instead of squared deviations, which describes mean absolute deviation rather than variance. Option D correctly squares each deviation before summing and then divides by n, which is the definition of population variance.

The mean equals 45 divided by 5 equals 9. The deviations are negative 4, negative 2, 0, 2, and 4. Squaring each gives 16, 4, 0, 4, and 16. The sum of squared deviations is 40. Dividing by n equals 5 gives population variance equals 40 divided by 5 equals 8.

Option A is correct: sample variance squares each deviation from the mean, sums them, and divides by n minus 1 applying Bessel's correction. Option B gives the population variance formula because it divides by n rather than n minus 1. Option C squares the raw values rather than the deviations from the mean. Option D takes a square root at the end, which would give a modified form of standard deviation rather than variance.

The answer is False. Sample variance divides by n minus 1 while population variance divides by n. Since n minus 1 is smaller than n, dividing the same sum of squared deviations by a smaller number produces a larger result. The sample variance is therefore always greater than or equal to the population variance for the same data set. The division by n minus 1 is an intentional upward adjustment called Bessel's correction, which compensates for the tendency of a sample to underestimate the true population spread.

The sum of squared deviations is 9 plus 1 plus 4 plus 16 equals 30. For population variance divide by n equals 4. Population variance equals 30 divided by 4 equals 7.5. When squared deviations are provided directly the summing and dividing by n steps are all that remain. Option C (8.0) would result from dividing by 3.75 rather than 4, and option A (5.5) would result from dividing by an incorrect denominator.

Computing variance for each: {5,5,5,5} has variance 0 since all values are identical. {2,4,6,8} has mean 5, squared deviations 9,1,1,9, and variance equals 20 divided by 4 equals 5. {3,4,5,6} has mean 4.5, squared deviations 2.25,0.25,0.25,2.25, and variance equals 5 divided by 4 equals 1.25. {1,1,9,9} has mean 5, squared deviations 16,16,16,16, and variance equals 64 divided by 4 equals 16. The data set {1,1,9,9} has the largest variance because its values are furthest from the mean.

Option A is true: when all values are identical every deviation equals 0, so every squared deviation equals 0, and the average of those squared deviations is 0. Option B is true: squaring any real number always produces a non-negative result, so the average of squared deviations can never be negative. Option C is incorrect: variance uses squared deviations while mean absolute deviation uses the absolute values of deviations — they are related measures of spread but are not equal and do not produce the same numerical result. Option D is true: multiplying every value by k multiplies each deviation by k, and squaring those deviations multiplies each by k squared, so the average of the squared deviations also increases by a factor of k squared.

When every value is multiplied by a constant k the variance is multiplied by k squared. Here k equals 3 so k squared equals 9 and the new variance is 9 times the original. This happens because each deviation from the mean is multiplied by k and when those deviations are squared the factor becomes k squared. For example if the original variance is 4 then after multiplying all values by 3 the new variance is 9 multiplied by 4 equals 36.

The answer is True. Standard deviation is defined as the square root of variance for both population and sample calculations. Population standard deviation equals the square root of population variance, and sample standard deviation equals the square root of sample variance. This means standard deviation and variance always describe the same spread but in different units: standard deviation is in the original data units while variance is in squared units, making standard deviation easier to interpret in context.

Variance measures how far data values are from the mean on average in squared units. A larger variance means the squared deviations are larger on average, indicating the values are more spread out from the mean. Variance is not affected by the number of values in the data set or the size of the mean itself. A data set with values clustered close to the mean will have small variance regardless of how many values it contains.

The mean equals 30 divided by 5 equals 6. The deviations are negative 4, negative 2, 0, 2, and 4. Squaring each gives 16, 4, 0, 4, and 16. The sum of squared deviations is 40. Dividing by n equals 5 gives population variance equals 40 divided by 5 equals 8.

The mean equals 40 divided by 5 equals 8. The deviations are negative 6, negative 3, 0, 3, and 6. Squaring each gives 36, 9, 0, 9, and 36. The sum of squared deviations is 90. For sample variance divide by n minus 1 equals 4. Sample variance equals 90 divided by 4 equals 22.5.

Standard deviation equals the square root of variance. The square root of 9 equals 3. Option D (81) results from squaring the variance rather than taking its square root. Option B (4.5) results from dividing the variance by 2 instead of taking the square root. Option A (2) has no correct derivation from a variance of 9. Only option C correctly applies the square root relationship.

The answer is False. A variance of 0 means every squared deviation equals 0, which means every deviation equals 0, which means every value equals the mean. If every value equals the mean then all values in the data set must be identical. It is impossible for a data set with variance 0 to contain any values that differ from one another. Variance of 0 is the definitive indicator that all data values are the same.

The mean equals 8 divided by 4 equals 2. The deviations are negative 1, negative 1, negative 1, and 3. Squaring each gives 1, 1, 1, and 9. The sum of squared deviations is 12. Dividing by n equals 4 gives population variance equals 12 divided by 4 equals 3. The single outlier value of 5 produces the largest squared deviation of 9 and is the main contributor to the variance in this data set.

Option A is correct: standard deviation is defined as the square root of variance. Option B is correct and equivalent: squaring both sides of option A gives variance equals standard deviation squared. Option C is incorrect and reverses the relationship — variance is not the square root of standard deviation. Option D is correct: if variance equals 0 then its square root equals 0, confirming standard deviation is also 0. A variance of 0 means all values are identical so there is no spread at all.

Squaring each deviation: negative 3 squared equals 9, negative 1 squared equals 1, 1 squared equals 1, and 3 squared equals 9. The sum of squared deviations is 9 plus 1 plus 1 plus 9 equals 20. Dividing by n equals 4 gives population variance equals 20 divided by 4 equals 5. When deviations are given directly the first two steps of the variance process are already complete, so only the squaring, summing, and dividing steps remain.

The mean equals 20 divided by 4 equals 5. The deviations are negative 2, negative 2, 2, and 2. Squaring each gives 4, 4, 4, and 4. The sum of squared deviations is 16. Dividing by n equals 4 gives population variance equals 16 divided by 4 equals 4. This data set has two values below the mean and two above, all equally distant from the mean of 5.

The answer is True. Adding a constant shifts every data value and the mean by the same amount. As a result each deviation (value minus mean) remains identical to what it was before the constant was added. Since the deviations are unchanged the squared deviations are unchanged and the variance stays the same. For example adding 10 to every value in {1, 2, 3} produces {11, 12, 13} which has the same variance as the original set.

The mean equals 24 divided by 3 equals 8. The deviations are negative 2, 0, and 2. Squaring each gives 4, 0, and 4. The sum of squared deviations is 8. For sample variance divide by n minus 1 equals 2. Sample variance equals 8 divided by 2 equals 4. The population variance for the same set would be 8 divided by 3 equals approximately 2.67, which is smaller than the sample variance.