What is the value calculated by adding all the values in a data set and dividing the sum by the number of values?

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The arithmetic mean is often known simply as the mean. It is an average, a measure of the centre of a set of data. The arithmetic mean is calculated by adding up all the values and dividing the sum by the total number of values.

For example, the mean of \(7\), \(4\), \(5\) and \(8\) is \(\frac{7+4+5+8}{4}=6\).

If the data values are \(x_1\), \(x_2\), …, \(x_n\), then we have \(\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i\), where \(\bar{x}\) is a symbol representing the mean of the \(x_i\) values.

This rearranges to give the useful result \[n\bar x = \sum_{i=1}^n x_i,\] that is, the arithmetic mean is the number \(\bar x\) for which having \(n\) copies of this number gives the same sum as the original data. So the sum of a set of numbers in some sense “averages” them.

If the data are grouped, with \(f_i\) occurrences of the value \(x_i\) for \(i=1\), \(2\), …, \(n\), then their mean is given by \[\bar{x}=\frac{\sum_{i=1}^n f_ix_i}{\sum_{i=1}^n f_i},\] where the numerator is the sum of all of the \(x_i\) values and the denominator is the total number of values.

The arithmetic mean is sensitive to outlier values.

The mean value of a function \(f(x)\) over the interval \(a\le x\le b\) is likewise the value \(M\) for which the constant function \(f(x)=M\) has the same “sum” as the original function. The “sum” of a function over an interval is the integral of the function, as shown in this sketch:

Thus the mean \(M\) is given by \(M(b-a)=\int_a^b f(x)\,dx\), so \[M=\frac{\int_a^b f(x)\,dx}{b-a}.\] The integral therefore “averages” the function.

Learning Outcomes

• Find the mean of a set of numbers

The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.

Suppose Ethan’s first three test scores were [latex]85,88,\text{and }94[/latex]. To find the mean score, he would add them and divide by [latex]3[/latex].

[latex]\begin{array}{}\\ {\Large\frac{85+88+94}{3}}=\\ {\Large\frac{267}{3}}=\\ 89\end{array}[/latex]

His mean test score is [latex]89[/latex] points.

The mean of a set of [latex]n[/latex] numbers is the arithmetic average of the numbers.

[latex]\text{mean}={\Large\frac{\text{sum of values in data set}}{n}}[/latex]

Calculate the mean of a set of numbers.

1. Write the formula for the mean
   [latex]\text{mean}={\Large\frac{\text{sum of values in data set}}{n}}[/latex]
2. Find the sum of all the values in the set. Write the sum in the numerator.
3. Count the number, [latex]n[/latex], of values in the set. Write this number in the denominator.
4. Simplify the fraction.
5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.

Find the mean of the numbers [latex]8,12,15,9,\text{ and }6[/latex].

Solution

The ages of the members of a family who got together for a birthday celebration were [latex]16,26,53,56,65,70,93,\text{ and }97[/latex] years. Find the mean age.

Did you notice that in the last example, while all the numbers were whole numbers, the mean was [latex]59.5[/latex], a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money, and it will make sense to report the mean in dollars and cents.

For the past four months, Daisy’s cell phone bills were [latex]\text{\$42.75},\text{\$50.12},\text{\$41.54},\text{\$48.15}[/latex]. Find the mean cost of Daisy’s cell phone bills.

In the next video we show an example of how to find the mean of a set of test scores.

Let’s go back to Mount Rival soccer tournament. Suppose that five teams were competing, each of them including 10 players for a total of 50 players. The number of goals scored by each player was compiled and results were summarized in the frequency table below. For example, we can see that eight players scored only one goal during the tournament. What is the average number of goals scored by the players during the tournament?

Table summary
This table displays the results of Number of players by the number of goals scored. The information is grouped by Number of goals scored (appearing as row headers), Number of players (appearing as column headers).

You first need to calculate the total number of goals scored. To do that, you take each observed value of the number of goals scored, which are values 0 to 6, and you multiply each value by the number of players:

0 × 2 + 1 × 8 + 2 × 14 + 3 × 12 + 4 × 8 + 5 × 4 + 6 × 2 = 136

Since there are 50 players, the average is 136 ÷ 50 = 2.72 goals per player.

Though we commonly use the word average in everyday life when discussing the number that’s the most “typical” or that’s “in the middle” of a group of values, more precise terms are used in math and statistics. Namely, the words mean, median, and mode each represent a different calculation or interpretation of which value in a data set is the most common or most representative of the set as a whole.

In this article, we’ll answer these questions and more:

• Is there a difference between mean and average?
• What’s the difference between mean, median, and mode?
• How do you find the mean, median, and mode?

You find the mean (informally called the average) by adding up all the numbers in a set and then dividing by how many values there are. When you arrange a set of values from smallest to largest, the median is the one in the middle. The mode is simply the value that occurs the most in the set.

mean vs. average

In math, the word mean refers to what’s informally called the average. They mean the same thing, but in the context of math and statistics, it’s better to use the word mean to distinguish from other things that might be casually referred to as “average” values in a general sense (meaning values that are the most representative or common within the set).

What is the mean, median, and mode?

The mean is the number you get by dividing the sum of a set of values by the number of values in the set.

In contrast, the median is the middle number in a set of values when those values are arranged from smallest to largest.

The mode of a set of values is the most frequently repeated value in the set.

To illustrate the difference, let’s look at a very simple example.

Here’s an example set of seven values: 2, 3, 3, 4, 6, 8, 9.

To find the mean: add up all the values (2+3+3+4+6+8+9=35) and then divide that total by the number of values (7), resulting in a mean of 5. This is what most people are referring to when they refer to the average of some set of numbers.

To find the median: find the value that’s sequentially in the middle. In a set of seven numbers arranged in increasing value, the median is the fourth number (since there are three before and three after). In this set (2, 3, 3, 4, 6, 8, 9), the median is 4. When a set has an even number of values, the median is the mean of the two middle values (in other words, you find the median by adding the two middle numbers together and dividing by two).

To find the mode: simply look to see which value shows up the most. In the example set, the mode is 3, since it occurs twice and all the other values occur only once.

Of course, this set of values is very simple compared to data sets you’re likely to encounter in real life. In cases when there is too much data to look at all at once, there are often special tools (such as those used in spreadsheet software) that you can use to determine the mean, median, and mode.