Why divide the sample variance by N-1?
- 1 Introduction
- 2 Minimum variance, unbiased estimators
- 3 Maximum Likelihood estimation
- 4 Estimating the variance if the mean is known
- 5 Estimating the variance if the mean is unknown
- 6 Conclusion
In this article, we will derive the well known formulas for calculating the mean and the variance of normally distributed data, in order to answer the question in the article’s title. However, for readers who are not interested in the ‘why’ of this question but only in the ‘when’, the answer is quite simple:
If you have to estimate both the mean and the variance of the data (which is typically the case), then divide by N-1, such that the variance is obtained as:
If, on the other hand, the mean of the true population is known such that only the variance needs to be estimated, then divide by N, such that the variance is obtained as:
Whereas the former is what you will typically need, an example of the latter would be the estimation of the spread of white Gaussian noise. Since the mean of white Gaussian noise is known to be zero, only the variance needs to be estimated in this case.
If data is normally distributed we can completely characterize it by its mean and its variance . The variance is the square of the standard deviation which represents the average deviation of each data point to the mean. In other words, the variance represents the spread of the data. For normally distributed data, 68.3% of the observations will have a value between and . This is illustrated by the following figure which shows a Gaussian density function with mean and variance :
Usually we do not have access to the complete population of the data. In the above example, we would typically have a few observations at our disposal but we do not have access to all possible observations that define the x-axis of the plot. For example, we might have the following set of observations:
|Observation ID||Observed Value|
If we now calculate the empirical mean by summing up all values and dividing by the number of observations, we have:
Usually we assume that the empirical mean is close to the actually unknown mean of the distribution, and thus assume that the observed data is sampled from a Gaussian distribution with mean . In this example, the actual mean of the distribution is 10, so the empirical mean indeed is close to the actual mean.
Again, we usually assume that this empirical variance is close to the real and unknown variance of underlying distribution. In this example, the real variance was 9, so indeed the empirical variance is close to the real variance.
The question at hand is now why the formulas used to calculate the empirical mean and the empirical variance are correct. In fact, another often used formula to calculate the variance, is defined as follows:
The only difference between equation (2) and (3) is that the former divides by N-1, whereas the latter divides by N. Both formulas are actually correct, but when to use which one depends on the situation.
In the following sections, we will completely derive the formulas that best approximate the unknown variance and mean of a normal distribution, given a few samples from this distribution. We will show in which cases to divide the variance by N and in which cases to normalize by N-1.
A formula that approximates a parameter (mean or variance) is called an estimator. In the following, we will denote the real and unknown parameters of the distribution by and . The estimators, e.g. the empirical average and empirical variance, are denoted as and .
To find the optimal estimators, we first need an analytical expression for the likelihood of observing a specific data point , given the fact that the population is normally distributed with a given mean and standard deviation . A normal distribution with known parameters is usually denoted as . The likelihood function is then:
To calculate the mean and variance, we obviously need more than one sample from this distribution. In the following, let vector be a vector that contains all the available samples (e.g. all the values from the example in table 1). If all these samples are statistically independent, we can write their joint likelihood function as the sum of all individual likelihoods:
Equation (6) will be important in the next sections and will be used to derive the well known expressions for the estimators of the mean and the variance of a Gaussian distribution.
Minimum variance, unbiased estimators
To determine if an estimator is a ‘good’ estimator, we first need to define what a ‘good’ estimator really is. The goodness of an estimator depends on two measures, namely its bias and its variance (yes, we will talk about the variance of the mean-estimator and the variance of the variance-estimator). Both measures are briefly discussed in this section.
Imagine that we could obtain different (disjoint) subsets of the complete population. In analogy to our previous example, imagine that, apart from the data in Table 1, we also have a Table 2 and a Table 3 with different observations. Then a good estimator for the mean, would be an estimator that on average would be equal to the real mean. Although we can live with the idea that the empirical mean from one subset of data is not equal to the real mean like in our example, a good estimator should make sure that the average of the estimated means from all subsets is equal to the real mean. This constraint is expressed mathematically by stating that the Expected Value of the estimator should equal the real parameter value:
If the above conditions hold, then the estimators are called ‘unbiased estimators’. If the conditions do not hold, the estimators are said to be ‘biased’, since on average they will either underestimate or overestimate the true value of the parameter.
Unbiased estimators guarantee that on average they yield an estimate that equals the real parameter. However, this does not mean that each estimate is a good estimate. For instance, if the real mean is 10, an unbiased estimator could estimate the mean as 50 on one population subset and as -30 on another subset. The expected value of the estimate would then indeed be 10, which equals the real parameter, but the quality of the estimator clearly also depends on the spread of each estimate. An estimator that yields the estimates (10, 15, 5, 12, 8) for five different subsets of the population is unbiased just like an estimator that yields the estimates (50, -30, 100, -90, 10). However, all estimates from the first estimator are closer to the true value than those from the second estimator.
Therefore, a good estimator not only has a low bias, but also yields a low variance. This variance is expressed as the mean squared error of the estimator:
A good estimator is therefore is a low bias, low variance estimator. The optimal estimator, if such estimator exists, is then the one that has no bias and a variance that is lower than any other possible estimator. Such an estimator is called the minimum variance, unbiased (MVU) estimator. In the next section, we will derive the analytical expressions for the mean and the variance estimators of a Gaussian distribution. We will show that the MVU estimator for the variance of a normal distribution requires us to divide the variance by under certain assumptions, and requires us to divide by N-1 if these assumptions do not hold.
Maximum Likelihood estimation
Although numerous techniques can be used to obtain an estimator of the parameters based on a subset of the population data, the simplest of all is probably the maximum likelihood approach.
The probability of observing was defined by equation (6) as . If we fix and in this function, while letting vary, we obtain the Gaussian distribution as plotted by figure 1. However, we could also choose a fixed and let and/or vary. For example, we can choose like in our previous example. We also choose a fixed , and we let vary. Figure 2 shows the plot of each different value of for the distribution with the proposed fixed and :
In the above figure, we calculated the likelihood by varying for a fixed . Each point in the resulting curve represents the likelihood that observation is a sample from a Gaussian distribution with parameter . The parameter value that corresponds to the highest likelihood is then most likely the parameter that defines the distribution our data originated from. Therefore, we can determine the optimal by finding the maximum in this likelihood curve. In this example, the maximum is at , such that the standard deviation is . Indeed if we would calculate the variance in the traditional way, with a given , we would find that it is equal to 7.8:
Therefore, the formula to compute the variance based on the sample data is simply derived by finding the peak of the maximum likelihood function. Furthermore, instead of fixing , we let both and vary at the same time. Finding both estimators then corresponds to finding the maximum in a two-dimensional likelihood function.
To find the maximum of a function, we simply set its derivative to zero. If we want to find the maximum of a function with two variables, we calculate the partial derivative towards each of these variables and set both to zero. In the following, let be the optimal estimator for the population mean as obtained using the maximum likelihood method, and let be the optimal estimator for the variance. To maximize the likelihood function we simply calculate its (partial) derivatives and set them to zero as follows:
In the following paragraphs we will use this technique to obtain the MVU estimators of both and . We consider two cases:
The first case assumes that the true mean of the distribution is known. Therefore, we only need to estimate the variance and the problem then corresponds to finding the maximum in a one-dimensional likelihood function, parameterized by . Although this situation does not occur often in practice, it definitely has practical applications. For instance, if we know that a signal (e.g. the color value of a pixel in an image) should have a specific value, but the signal has been polluted by white noise (Gaussian noise with zero mean), then the mean of the distribution is known and we only need to estimate the variance.
The second case deals with the situation where both the true mean and the true variance are unknown. This is the case you would encounter most and where you would obtain an estimate of the mean and the variance based on your sample data.
In the next paragraphs we will show that each case results in a different MVU estimator. More specific, the first case requires the variance estimator to be normalized by to be MVU, whereas the second case requires division by to be MVU.
Estimating the variance if the mean is known
If the true mean of the distribution is known, then the likelihood function is only parameterized on . Obtaining the maximum likelihood estimator then corresponds to solving:
However, calculating the derivative of , defined by equation (6) is rather involved due to the exponent in the function. In fact, it is much easier to maximize the log-likelihood function instead of the likelihood function itself. Since the logarithm is a monotonous function, the maximum will be the same. Therefore, we solve the following problem instead:
In the following we set to obtain a simpler notation. To find the maximum of the log-likelihood function, we simply calculate the derivative of the logarithm of equation (6) and set it to zero:
Note that this maximum likelihood estimator for is indeed the traditional formula to calculate the variance of normal data. The normalization factor is .
However, the maximum likelihood method does not guarantee to deliver an unbiased estimator. On the other hand, if the obtained estimator is unbiased, then the maximum likelihood method does guarantee that the estimator is also minimum variance and thus MVU. Therefore, we need to check if the estimator in equation (10) is unbiassed.
To do this, we plug equation (10) into and write:
Furthermore, an important property of variance is that the true variance can be written as such that . Using this property in the above equation yields:
Since , the condition shown by equation (7) holds, and therefore the obtained estimator for the variance of the data is unbiassed. Furthermore, because the maximum likelihood method guarantees that an unbiased estimator is also minimum variance (MVU), this means that no other estimator exists that can do better than the one obtained here.
Therefore, we have to divide by instead of while calculating the variance of normally distributed data, if the true mean of the underlying distribution is known.
Estimating the variance if the mean is unknown
In the previous section, the true mean of the distribution was known, such that we only had to find an estimator for the variance of the data. However, if the true mean is not known, then an estimator has to be found for the mean too. Furthermore, this mean estimate is used by the variance estimator. As a result, we will show that the earlier obtained estimator for the variance is no longer unbiassed. Furthermore, we will show that we can ‘unbias’ the estimator in this case by dividing by instead of by , which slightly increases the variance of the estimator.
As before, we use the maximum likelihood method to obtain the estimators based on the log-likelihood function. We first find the ML estimator for :
If , then it is clear that the above equation only has a solution if:
Note that indeed this is the well known formula to calculate the mean of a distribution. Although we all knew this formula, we now proved that it is the maximum likelihood estimator for the true and unknown mean of a normal distribution. For now, we will just assume that the estimator that we found earlier for the variance , defined by equation (10), is still the MVU variance estimator. In the next section however, we will show that this estimator is no longer unbiased now.
To check if the estimator for the true mean is unbiassed, we have to make sure that the condition of equation (7) holds:
Since , this means that the obtained estimator for the mean of the distribution is unbiassed. Since the maximum likelihood method guarantees to deliver the minimum variance estimator if the estimator is unbiassed, we proved that is the MVU estimator of the mean.
To check if the earlier found estimator for the variance is still unbiassed if it is based on the empirical mean instead of the true mean , we simply plug the obtained estimator into the earlier derived estimator of equation (10):
To check if the estimator is still unbiased, we now need to check again if the condition of equation (7) holds:
As mentioned in the previous section, an important property of variance is that the true variance can be written as such that . Using this property in the above equation yields:
Since clearly , this shows that estimator for the variance of the distribution is no longer unbiassed. In fact, this estimator on average overestimates the true variance with a factor . As the number of samples approaches infinity (), this bias converges to zero. For small sample sets however, the bias is signification and should be eliminated.
Fixing the bias
Since the bias is merely a factor, we can eliminate it by scaling the biased estimator defined by equation (10) by the inverse of the bias. We therefore define a new, unbiased estimate as follows:
This estimator is now unbiassed and indeed resembles the traditional formula to calculate the variance, where we divide by instead of . However, note that the resulting estimator is no longer the minimum variance estimator, but it is the estimator with the minimum variance amongst all unbiased estimators. If we divide by , then the estimator is biassed, and if we divide by , the estimator is not the minimum variance estimator. However, in general having a biased estimator is much worse than having a slightly higher variance estimator. Therefore, if the mean of the population is unknown, division by should be used instead of division by .
In this article, we showed where the usual formulas for calculating the mean and the variance of normally distributed data come from. Furthermore, we have proven that the normalization factor in the variance estimator formula should be if the true mean of the population is known, and should be if the mean itself also has to be estimated.