Estimating Variance

Section 8.3 Estimating Variance

As another example, suppose we want to estimate variance in the penguins’ weights. In Chapter 1, we saw that there are two ways to compute the variance of a sample. I promised to explain the difference later -- and later is now.

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The reason there are two ways to compute the variance of a sample is that one is a biased estimator of the population variance, and the other is unbiased. The following function computes the biased estimator, which is the sum of the squared deviations divided by n.

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Listing 8.3.1. Python Code

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def biased_var(xs):
    # Compute variance with n in the denominator
    n = len(xs)
    deviations = xs - np.mean(xs)
    return np.sum(deviations**2) / n

To test it, we’ll simulate many samples with size 10, compute the biased variance of each sample, and then compute the average of the variances.

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Listing 8.3.2. Python Code

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$ biased_vars = [biased_var(make_sample(n=10)) for i in range(10001)]
np.mean(biased_vars)
np.float64(0.19049277659404473)

The result is about 0.19, but in this case, we know that the actual population variance is about 0.21, so this version of the sample variance is too low on average -- which confirms that it is biased.

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Listing 8.3.3. Python Code

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$ actual_var = sigma**2
actual_var
0.2116

The following function computes the unbiased estimator, which is the sum of the squared deviations divided by n-1.

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Listing 8.3.4. Python Code

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def unbiased_var(xs):
    # Compute variance with n-1 in the denominator
    n = len(xs)
    deviations = xs - np.mean(xs)
    return np.sum(deviations**2) / (n - 1)

We can test it by generating many samples and computing the unbiased variance for each one.

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Listing 8.3.5. Python Code

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$ unbiased_vars = [unbiased_var(make_sample(n=10)) for i in range(10001)]
np.mean(unbiased_vars)
np.float64(0.21159109492300626)

The average of the unbiased sample variances is very close to the actual value -- which is what we expect if it is unbiased.

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With sample size 10, the difference between the biased and unbiased estimators is about 10%, which might be non-negligible. With sample size 100, the difference is only 1%, which is small enough that it probably doesn’t matter in practice.

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Listing 8.3.6. Python Code

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$ n = 10
1 - (n - 1) / n
0.09999999999999998

Listing 8.3.7. Python Code

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$ n = 100
1 - (n - 1) / n
0.010000000000000009

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