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Use sample data to estimate a mean, a variance and a standard deviation.

FRM® Part I · Quantitative Analysis · Sample Moments · FRM1.QA.5.a

The idea

The sample mean is the sum of the observations divided by n. The sample variance is the sum of squared deviations from the sample mean divided by n − 1: Variance = Σ(x − x̄)² ÷ (n − 1). The sample standard deviation is its square root. The divisor n − 1 corrects for the fact that deviations are measured from a mean fitted to the same data, which makes them slightly too small. With n as the divisor, the variance would be biased low. If you only have the sum of x and the sum of x², the centred sum of squares is: SS = Σx² − n × x̄². A common slip is dividing by n, or forgetting that the standard deviation, unlike the variance, is still slightly biased downward even with n − 1.

Formulas

Sample variance

s² = Σ(xᵢ − x̄)² ÷ (n − 1)

xᵢ
each observation
x̄
the sample mean
n
the number of observations

Dividing by n − 1 removes the bias from measuring deviations around a fitted mean. The square root is still slightly biased low.

Centred sum of squares from raw sums

SS = Σx² − n × x̄²

x
each observation
n
the number of observations
x̄
the sample mean, Σx ÷ n

Use when only the sum and the sum of squares are given. Divide by n − 1 to get the sample variance.

A practice question

From the question bank, with the reason each wrong option is wrong.

A risk analyst has five monthly returns of 2%, -1%, 4%, 0% and 3%. What is the sample mean return?

  1. 1.60% - correct
  2. 8.00%Why not: Adds the observations but forgets to divide by n.
  3. 2.00%Why not: Reports the median (the middle ordered value) instead of the mean.
  4. 1.33%Why not: Divides by n + 1 rather than n.

The sample mean is the sum divided by the number of observations: (2 - 1 + 4 + 0 + 3) / 5 = 1.60%.

A harder one

Six daily P&L returns (in %) are 1.2, -0.8, 0.5, 2.1, -1.5 and 0.3. What is the sample standard deviation, using the n - 1 divisor, closest to?

  1. 1.19%Why not: Uses the divisor n instead of n - 1.
  2. 1.71%Why not: Reports the sample variance and forgets to take the square root.
  3. 1.31% - correct
  4. 1.35%Why not: Does not subtract the sample mean: sums the squared returns and divides by n - 1.

Mean = 0.30. Sum of squared deviations = 8.540. Variance = 8.540/5 = 1.708. Standard deviation = sqrt(1.708) = 1.31%.

The hardest tier

Ten observations have sample mean 2.0 and sample variance 4.0 (n - 1 divisor). An eleventh observation of 5.0 arrives. What is the updated sample variance (n - 1 divisor), closest to?

  1. 4.42 - correct
  2. 4.50Why not: Measures the new point from the old mean only and ignores that the mean itself moves.
  3. 4.02Why not: Gets the new sum of squares right but divides by n + 1 = 11 instead of 10.
  4. 4.34Why not: Adds the new point's squared deviation from the new mean but leaves the old sum of squares measured about the old mean.

New mean = (10 x 2 + 5)/11 = 2.2727. Old sum of squares = 9 x 4 = 36. Updating: 36 + (5 - 2)(5 - 2.2727) = 44.182. Divide by 10: 4.418.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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