How to find the mean score of a set of numbers

People often talk about finding the average of a set of numbers, such as household income, test scores, or traffic accidents. However, in statistics, there are several types of averages. The mean score is often used and differs from other averages, such as median and mode.

A mean score, also known as the arithmetic average, is calculated by adding up all the values in a set of numbers and then dividing by the total number of values. The mean score is often used as a measure of central tendency, which represents the typical or most representative value in a dataset.

How to find a mean score

Let's take an example to see how a mean score is found. Suppose you want to find the mean score of students who took a test. Assume that there are a total of 10 test scores:

• 75

• 62

• 85

• 58

• 91

• 73

• 98

• 69

• 88

• 90

Adding up these numbers, we get a total of 789. Dividing this by 10, we get 78.9, which can be rounded up to 80.

For another example, suppose a theater or stadium wanted to calculate the mean attendance over three events.

• 10,575

• 7,430

• 14,186

The total for the three events is 32,191. Dividing this total by 3, we get 10,730. 

What are mean scores and standard scores?

Standard scores are scores with the same mean and standard deviation. Standard deviation is used to account for variations from the mean. For example, the mean test score for students is 95%. However, the standard score may be lower if we account for the standard deviation created by a few students who scored 98%–100%.

The arithmetic mean, as defined above, is the most commonly used type of mean score. There are also others.

Weighted mean

Often used in statistics, a weighted mean is for when certain variables are more important (or carry more "weight") than others. 

It's calculated by multiplying each value in a set by its assigned weight and then adding up the totals.

As is often the case, not all tests are equal in terms of an overall study grade. For example, suppose you want to calculate a weighted mean for a student’s performance on three exams. Suppose the first two exams contribute 25% of the student's grade while the third contributes 50%. The student then gets the following grades on each exam.

• Exam 1: 80%

• Exam 2: 100%

• Exam 3: 90% 

Multiply the grades by the weight. 

• Exam 1: .25 x 80 = 20

• Exam 2: .25 x 100 = 25

• Exam 3: .50 x 90 = 45

Now we would add up the totals: 20 + 25 + 45 = 90. The weighted average is 90 for the student’s overall grade.

Geometric mean

Geometric means are often used to track the performance of investments, profits, or economic variables such as the inflation rate. It allows investors and analysts to identify the long-term value of an investment.

The geometric mean is calculated by multiplying all values and then identifying the nth root of the product, where n is the number of values.

If you wanted to find the geometric mean between two numbers, such as 4 and 9, you would first multiply them, giving you 36. You would then take the square root of 36, which is 6, which is the geometric mean.

There are certain limitations when it comes to geometric means. It can only be used for positive numbers and cannot be used if losses are involved. If any of the variables is 0, the geometric average will also be 0.

Harmonic mean

Harmonic means are often used to compare companies’ price-earning ratios (P/E). It is especially useful when there is a need to give greater weight to smaller variables. 

The harmonic mean is found by dividing the number of items in a series by the sum of each number's reciprocal. The reciprocal is found by dividing one by that number. For example, the reciprocal of four is 1/4.

If we want to find the harmonic mean of the values 4, 2, and 4, we'd add the reciprocal values. 

.25 + .5 + .25 = 1

Since there are three items, we get 3/1 = 1.

When people speak of averages, they may be referring to other frequently-used metrics such as:

• Median — The middle value of a series. For example, in the series of numbers 1 through 10, the median is 5.5.

• Mode — The number that occurs most frequently in a data set. For example, if you have a set of exam scores of 63, 70, 94, 59, 84, 70, and 88, the mode would be 70. Mode is most useful when you have a large data set and want to identify the most common value. 

• Range — The largest number minus the smallest number in a series. For example, in the series of numbers 2, 8, 12, 24, and 45, the range is 45-2 = 43.

Mean scores are helpful for identifying typical or frequently occurring scores or values. Teachers and school administrators want to know students' mean scores to track learning and progress. Financial advisors need to be aware of how assets perform over time, so mean scores help track the ups and downs of the market.

When analyzing data, it helps to be aware of factors that can affect your results. These may include any of the following.

Outliers

These are values that are much higher or lower than the average and can affect the mean. For example, a student who scores much higher or lower than their classmates will raise or lower the mean. A very wealthy individual moving into a small town will artificially inflate the area’s mean income.

Seasonal factors

In business and finance, seasonal factors such as weather, holidays, and typical consumer habits can affect data. For example, data might show that mean retail profits rose in December, but this is likely to be a normal holiday fluctuation and not necessarily indicative of an economic upturn.

Differences in groups

If data is collected from a broad range of sets, researchers may come to misleading conclusions. As an article in Towards Data Science describes, the divorce rate in the United States is commonly cited to be around 50%. This, however, doesn't account for very large differences based on factors such as age and education.

There are advantages and disadvantages to using mean scores. The main advantage of mean scores is that they provide a clear idea of the central or most common tendency. The downside is that outliers can distort the mean. You can avoid this by calculating the standard score. Using other averages, such as the median, can also give you a more complete understanding of the data.