How Do You Find the Mean? A Step-by-Step Guide

The mean, commonly known as the average, is a frequently used measure of central tendency in statistics. But what exactly is the mean and how do you calculate it? This comprehensive guide will walk you through everything you need to know to find the mean of any dataset.

What is the Mean?

The mean, also called the arithmetic mean or average, is the sum of all values in a dataset divided by the total number of values. It gives you a single number that is representative of the middle of the dataset. The mean is the most commonly used measure of central tendency because it takes into account every value in your data.

Here is the formula for calculating the mean of a sample

![Sample Mean Formula][]

Where:

• x̄ = mean of sample
• Σx = sum of all values in sample
• n = total number of values in sample

And here is the formula for the population mean ![Population Mean Formula][]

Where:

• μ = mean of population
• ΣX = sum of all values in population
• N = total number of values in population

The only difference between the formulas is that we use lowercase (x) for sample attributes and uppercase (X) for population attributes,

When to Use the Mean

The mean is the best measure of central tendency to use when:

• You have continuous, quantitative data. The mean requires numerical values that can be added together.
• Your data has a normal distribution. In a normal distribution, the mean, median, and mode are all equal.
• You want to use all values in your dataset. The mean takes into account every data point.
• You have no significant outliers. Extreme outliers can skew the mean by pulling it towards them.

Steps to Calculate the Mean

Follow these two simple steps to find the mean of any dataset:

1. Add Up All the Values

First, add up all of the values in your dataset. Be sure to keep track of negative signs if you have negative numbers.

For example, if your dataset is:

{2, 7, -3, 10, 7}

You would calculate:

2 + 7 + (-3) + 10 + 7 = 23

2. Divide the Sum by Number of Values

Next, divide the sum you just calculated by the total number of values in the dataset.

Using our example, there are 5 values in the dataset.

So we would divide the sum (23) by the number of values (5):

23/5 = 4.6

The mean of this dataset is 4.6.

And that’s it! By following these two steps you can easily calculate the mean of any set of numbers.

Calculating the Mean by Hand

Let’s go through an example of calculating the mean of a dataset by hand:

The number of hours spent studying last week by 8 college students are:

{3, 5, 2, 4, 5, 8, 10, 6}

Step 1) Add up all the values:

3 + 5 + 2 + 4 + 5 + 8 + 10 + 6 = 43

Step 2) Divide the sum by the total number of values (n=8):

43/8 = 5.375

The mean number of hours spent studying is 5.375

Using a Calculator to Find the Mean

For larger datasets, it’s easier to use a calculator rather than adding up all the values by hand. Here are the steps:

1. Enter all the values from your dataset into the calculator.

2. Press the “sum” or “total” button on your calculator to add them up.

3. Divide the sum by the number of values in your dataset.

4. The result is the mean!

Many statistics calculators and spreadsheet programs like Excel have built-in functions to calculate the mean automatically. But it’s good to know how to do it manually as well.

The Effect of Outliers on the Mean

One important thing to understand about the mean is that it is sensitive to outliers. Outliers are extreme values that are much higher or lower than the other data points. Just one or two outliers can skew the mean by pulling it towards them.

Let’s see an example. Here is a dataset of daily temperatures:

{68, 65, 67, 71, 78, 62, 60}

The mean temperature is 67.4 ̊F

Now let’s add an outlier value of 95 ̊F to the dataset:

{68, 65, 67, 71, 78, 62, 60, 95}

The new mean temperature is now 71.1 ̊F

See how the outlier raised the mean substantially? The median or mode would likely better represent the central tendency of this skewed dataset.

Mean, Median, and Mode

The mean, median, and mode are the three main measures of central tendency in statistics. Here’s a quick overview:

• Mean – The average, calculated by summing all values and dividing by the number of values.
• Median – The middle value, found by ordering the dataset and selecting the value in the center.
• Mode – The most frequently occurring value in the dataset.

Choosing which one to use depends on the shape of your data distribution and whether you have outliers. The mean is best for symmetric, normal distributions without outliers. The median or mode are better for skewed distributions.

Mean For Grouped Frequency Tables

When you have data organized into groups with frequencies rather than raw scores, you need to use a modified formula to calculate the mean.

Here is the formula for mean of grouped frequencies:![Mean of Grouped Frequencies Formula][]

Where:

• x̄ = mean
• xi = midpoint of each class
• fi = frequency of each class

Let’s look at an example:![Grouped Frequency Table Example][]

To find the mean:

• Calculate the midpoint for each class by averaging the lower and upper limits
• Multiply each midpoint by the class frequency
• Add up the products
• Divide the sum by the total frequency (n)

Using this process gives us a mean of 67.7 for this data.

Weighted Mean

In some cases, you may want to give certain values in your dataset more influence over the mean. This is called a weighted mean.

The formula for weighted mean is:![Weighted Mean Formula][]

Where:

• x̄w = weighted mean
• wi = weight of each value
• xi = each individual value

Weights are applied by multiplying each value by its assigned weight. Higher weighted values have more impact on shifting the mean.

Let’s say you survey both students and faculty about how many events they attend on campus each semester. You want to give more weight to the faculty responses. You assign faculty a weight of 2 and students a weight of 1.

Here is the data:

| Group | Value | Weight |
|