Mean Calculator

Mean calculator at a glance#

What is the mean?#

The mean is calculated by summing up all the values and then dividing by the number of those values. It's often called the arithmetic mean and is widely known as the average. In statistics, the mean of a sample is denoted as x̄, while the mean of an entire population is represented by μ.

What is the mean of 350, 275, 400, 375, 450 and 370?#

The mean of these numbers is 370. If you add up all six values, you get 2,220, and when you divide 2,220 by 6, you arrive at 370. In this case, the mean is one of the values in the set, but that's not always how it turns out.

How do I use the mean calculator?#

Enter at least two numbers into the box above, using commas or spaces to separate them. The calculator will display the mean, the count (n), and the total sum of the values. The results are rounded to five decimal places.

The mean formula and its symbols#

For the sample mean, the formula is x̄ = Σx ÷ n. The symbol Σ (sigma) means "add up", so Σx represents the total sum of all values x, and n indicates how many values there are. For the population mean, the same arithmetic applies: μ = Σx ÷ N, where N is the total size of the population. In OpenStax Introductory Statistics, this notation is used, with x̄ pronounced as "x-bar" and μ as "mew".

For the data set mentioned:

• Σx = 350 + 275 + 400 + 375 + 450 + 370 = 2,220
• n = 6
• x̄ = 2,220 ÷ 6 = 370

Population mean vs sample mean#

The calculation method remains the same, but what the numbers stand for changes. When all six values are from the entire group of interest, like the scores of every person on a team, their mean is known as the population mean, denoted by μ. If they're taken from a larger group, the mean is referred to as the sample mean, represented by x̄, and it acts as an estimate of the population mean you haven't measured.

This is why this calculator provides a single answer for both means. The distinction becomes important when considering spread. For a sample, variance and standard deviation divide by n − 1 rather than N. Both the variance calculator and the standard deviation calculator cover the sample and population versions.

The mean as a balance point#

To find deviations, subtract the mean from each value. For our data set, with a mean of 370, the deviations are −20, −95, 30, 5, 80, and 0. The sum of the negative deviations is −115, and the sum of the positive deviations is 115, so the deviations total zero.

This principle applies to all data sets, making the mean the balance point of the data. The values work like equal weights on a ruler. The ruler balances at 370, even though more values are above it than below it, because the value of 275 is positioned far on the left.

How an outlier pulls the mean#

Every value contributes to the mean, so a single extreme value can significantly shift it. Imagine if the value 450 in our data set was mistakenly entered as 1,450. This mistake causes the sum to rise to 3,220, giving a mean of 3,220 ÷ 6 = 536.67. This mean is higher than five of the six values entered, meaning it no longer accurately represents a typical data point at all.

The median, however, doesn't change. In order, the values are 275, 350, 370, 375, 400, and 1,450. The median remains the average of the two middle numbers, 370 and 375, which is still 372.5, unchanged by the error. As highlighted by the NIST Engineering Statistics Handbook, the median generally gives a better estimate of the center than the mean when data includes extreme outliers, because it depends on the order of values rather than their magnitude.

If there's a notable discrepancy between the mean and median in your data, check for errors or unusual values before you finalize the mean. Then decide which statistic fits your question.

Mean vs median vs mode#

The mean, median, and mode each offer a different perspective on the center of a data set. The mean takes all values into account, the median identifies the middle value in sorted data, and the mode points to the most frequently occurring value. In our case with six values, the mean is 370, the median is 372.5, and there's no mode since no value repeats. The mean, median and mode calculator computes all three, along with the range and quartiles.

Means people often look up#

These data sets come from real questions. Each mean is calculated by dividing the sum by the count.

Sources#

• OpenStax, Introductory Statistics 2e, section 2.5: Measures of the Center of the Data
• NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.1: Measures of Location

Frequently asked questions#

What does x̄ mean in statistics?#

x̄, read as "x-bar", represents the sample mean, the average of the values you collected. The Greek letter μ denotes the population mean. Both are determined by summing the values and dividing by their number.

If x̄ = 75.13, what is the mean?#

The mean is 75.13. If you're given n = 60, x̄ = 75.13, and a variance of 15.15, you've already been provided the mean because x̄ is the sample mean. The count and variance describe the same data further.

Is the mean the same as the average?#

Yes, in common terms, the average refers to the arithmetic mean. The average calculator gives the same number, along with the median, range, and several other outcomes for a single list.

What is the mean deviation?#

The mean absolute deviation is the average distance of the values from the mean. In our data set, the distances from 370 are 20, 95, 30, 5, 80, and 0, which total 230, so the mean absolute deviation is 230 ÷ 6 = 38.33. This calculator doesn't compute it, but the steps are easy to follow by hand.

Does adding a value equal to the mean change the mean?#

No, it doesn't. Adding 370 to our six values gives a sum of 2,590 and a count of 7, making 2,590 ÷ 7 still 370. Introducing a new value above the mean will increase it, while a value below will decrease it.

Why does the calculator need at least two numbers?#

If you have only one number, the mean is simply that number, so there's nothing to calculate. When there are two or more values, the calculator sums them, counts them, and then divides.