In statistics, the symbols and formulas for basic concepts such as the mean provide a foundational understanding of data analysis. Understanding the mean involves more than just knowing how to calculate an average; it’s about recognizing the nuances that differentiate a population mean from a sample mean. This distinction is crucial in statistical analysis, as the approach and symbol used for each vary (mu vs. x bar).
This seemingly simple measure holds significant weight in interpreting sets of numbers, whether in academic research, business analytics, or even everyday data comprehension. The mean, commonly known as the arithmetic average, is one of the most fundamental concepts in statistics. As a measure of central tendency, it answers the question, where do most values fall?
In statistics, we work with population and sample means, each having its own symbol and formula. This post guides you through these symbols’ meanings and formulas as they apply to population and samples. Learn how to tell them apart!
Learn about Measures of Central Tendency: Mean, Median, and Mode.
Population Mean Symbol — μ or mu
The Greek letter µ (mu) is the symbol for a population mean. Statisticians frequently use Greek letters for measures of entire populations. We also refer to these population measures as parameters.
Suppose we measure the heights of the entire population of adult male basketball players in the United States and calculate an average height of 6 feet (1.83m). Because it’s a population mean, we use the mu symbol: µ = 6.
However, it’s usually impossible to measure an entire population. So, we’re often left with a sample mean. A sample is a subset of a population.
Sample Mean Symbol — x̅ or x bar
The symbol for a sample mean is x̅, which you pronounce as X-bar. This type of average can be less useful because it finds only the typical height of a particular sample.
However, if we collect a random sample, we can use x̅ to estimate µ. The process allows us to use the sample to estimate the properties of a population. That’s inferential statistics and it greatly extends the uses for sample means! Learn about the Differences between Descriptive and Inferential Statistics.
For example, imagine we draw a random sample from the population of basketball players and measure their heights. Then, we calculate that the average height of this subset of players is 6 feet. Because it’s a sample mean, we use the x bar symbol: x̅ = 6.
We know the average height of our specific sample of players is 6 feet.
Fortunately, we collected a random sample. Hence, this average is an unbiased estimate of the population mean for all basketball players. An unbiased x bar means that the estimate is equally likely to be too high or too low. In other words, it tends to be correct on average!
So, our x̅ = 6 serves as an estimate of the population mean: µ ≈ 6.
Learn more about Population vs. Sample and Parameter vs. Statistic.
Population and Sample Mean Formulas
The mean formula is simple yet powerful. Calculate the average by adding all the values in a dataset and dividing this total by the number of values.
Regarding the population and sample mean formulas, both follow a similar mathematical approach, with a subtle yet crucial differences.
Population Formula
Use the following formula to calculate the population mean. Again, notice that the µ mu symbol represents the population mean.
In this equation, the numerator sums all values in the population. That’s the ∑X in the numerator. In the denominator, N is the total number of values in the population. This formula is comprehensive, encompassing every single data point in the population.
In short, sum all values in the population and divide by the number of values in the population.
Easier said than done in most cases!
Sample Formula
Conversely, calculate the sample mean using the following formula, represented by the x̅ (x bar) symbol.
This formula is structurally similar to that of the population mean. We’re still summing all the values but only those in our subset. Additionally, notice the lowercase n in the denominator. This n represents the number of values in the sample rather than the population.
So, we sum all values in our subset and divide by the sample size.
Much easier!
Symbol and Formula Differences
In summary, the key differences between the two mean formulas are µ vs. x̅ (mu vs. x bar symbols) and N vs. n. In each case, the former relates to the population, while the latter is for the sample mean formula.
Summing up values and dividing by the number of items is consistent in both formulas. However, you’ll need to know whether you’re summing all values in the population or sample. And whether you divide by the total population size (N) or the sample size (n).
These distinctions between mu and x bar are crucial as they reflect the scope of the data you are analyzing – whether it’s the entire population or just a sample.
Crucially, calculating the population mean µ is generally impossible. However, you can use the sample mean x̅ to estimate it when using random samples. This point is where hypothesis testing enters the picture because it uses samples to infer properties of the population.
Why don’t we have degrees of freedom in mean while we consider degree of freedom in Variance (SD)
Hi Kishore,
When calculating the sample mean, we use all available data points without any intermediate estimates. Each data point contributes directly to the mean, and there’s no need to “adjust” for any other calculations.
In contrast, variance (or standard deviation) involves an additional step: we first calculate the mean, then measure each data point’s deviation from that mean. Since the mean is already an estimate based on our data, we “lose” one degree of freedom because we’re not entirely free to vary all observations independently—one value is essentially fixed by the mean. Therefore, when estimating variance from a sample, we adjust with n−1 degrees of freedom to make the estimate unbiased.
how to calculate the “global” mean when the subsets have a different amount of data points, e.g., subset 1 has 200 observations and subset 2 has 800 observations.
My understanding is that you cannot take the mean of subset 1, the mean of subset 2 and then add them together. What I have been doing is concatenating both sets of data and then calculating the mean of the combined data set, however, what is the formula for global mean when each subset has a different amount of observations.
Thanks
Hi Alan,
You’re correct that you can’t take the mean of the subsets when the subsets have different sample sizes. Instead, you need to calculate a weighted mean. Click the link to see how. That’ll give the subsets different weights based on their respective sizes.
I’m a noob so my understanding is likely incorrect, but you seem to be describing the “average”. I thought the “mean” was a value midway between all the values i.e. where half of the “mean” are greater than and half are less than the “mean” value.
Hi Wink!
The arithmetic average and mean are synonyms. In statistics, we usually refer to it as the mean rather than the average.
The median is the point where half the values are above and half below. That’s a different measure of central tendency. Click the link to learn more about them!
Whoops, my mistake, sorry!
No worries at all! Very easy to confuse them!