Average Calculator
The online average calculator makes it easy to find the average for any data set. You can type, copy, and paste your data into the data box. Make sure to separate each data point with a comma. Then, click the "Calculate" button.
The average calculator will show you the average (arithmetic mean), calculation steps, & other related statistics for the data set.
The Average
The average is defined as the mean of the values in a data set. All the values in the data set are used to calculate the average. Therefore, it represents the entire data set. The average is regarded as one of the most important central tendency or summary measures.
The simple arithmetic mean is the most common average. However, there are several kinds of averages, including the geometric mean, weighted average, combined arithmetic average, harmonic mean, and so forth.
The average of a population is represented by μ (Mu) and the average of a sample is represented by X̄ (X bar).
Simple Average
The simple average is calculated by dividing the data set's values by the total number of data items. The simple average is sometimes referred to as the mean, the arithmetic mean, and the average.
To calculate the average of a population, we can use the formula below.
μ = Sum of the data set's values / Total number of data values in the population = ΣX / N
To calculate the average of a sample, we can use the below formula:
X̄ = Sum of the data set's values / Total number of data values in the sample = ΣX/n
Let's learn the average using the below example.
Example:
Jasmine's scores for seven subjects from the previous semester are displayed in the table below. What is the average of Jasmine's previous semester subject scores?
| Subject |
Score |
| Management |
84 |
| Communication |
90 |
| Accounting |
75 |
| Economics |
60 |
| Business Statistics |
85 |
| International studies |
92 |
| Mathematics |
81 |
Solution:
The average score = ΣX / N = (84 + 90 + 75 + 60 + 85 + 92 + 81) / 7 = 567 / 7 = 81
The average is a concept everyone is familiar with. The average income, the average cost of production, average pricing, average score, average fuel consumption, etc., are a few examples you may have heard often. Even in everyday life, the simple average is a standard computation. The simple average or the simple arithmetic mean is also known as the ideal average.
In some situations, however, we use other measures of central tendency. Let's take a look at them.
Geometric Mean
The arithmetic mean is not an appropriate measurement when determining the average growth rate of a value over time. The geometric mean, which is often used in accounting and finance, such as in calculating compound interest, is a much better indicator for such calculations. This is because the growth rate is multiplicative rather than additive.
The geometric mean of your data set is defined as the nth root of the product of n items. It is calculated by multiplying each value together and then calculating the nth root of the product, where n is the number of items in the dataset. The geometric mean is helpful when averaging ratios, percentages, and growth rates.
Geometric Mean = n√(x₁×x₂×x₃×…×xₙ) = (x₁×x₂×x₃×…×xₙ)^(1/n)
We will find the Geometric Mean of the previous example.
Geometric Mean = ⁷√(84×90×75×60×85×92×81) = 80.31
The Geometric Mean is always equal to or lower than the simple average (arithmetic mean).
In our example,
Geometric Mean ≤ The average
80.31 < 81
You can use the average calculator to determine more than just the arithmetic mean. You can also use it to obtain your data set's Geometric Mean.
Weighted Average
In the simple arithmetic mean, all values have the same weight or importance. But in some cases we cannot apply the same level of importance to every value in our dataset.
In our example, we calculated the average by summing up all the scores and dividing by the total number of subjects. We haven't considered the relative importance of each subject.
The weighted average must be used when we need to consider the relative importance of each item of our data set when calculating the average. The weighted average is calculated by dividing the weighted values by the total of the weights. The data value multiplied by the relevant weight is the weighted value.
We can use the below formula to find the weighted average.
The weighted average = The sum of the weighted values / The sum of the weights = ΣWX / ΣW
Example:
Assume that each of the subjects in the previous example has a different weight. So, the updated data table for Jasmine's score in 7 subjects of the prior semester is as follows.
Weighted average of Jasmine's scores from the previous semester:
| Subject |
Score |
Weight |
| Management |
84 |
3 |
| Communication |
90 |
2 |
| Accounting |
75 |
4 |
| Economics |
60 |
3 |
| Business Statistics |
85 |
3 |
| International studies |
92 |
2 |
| Mathematics |
81 |
3 |
Solution:
The weighted average score = ΣWX / ΣW = (84×3+90×2+75×4+60×3+85×3+92×2+81×3)/(3+2+4+3+3+2+3) = (252+180+300+180+255+184+243)/20 = 1594/20 = 79.7
The Median
The median is the midway value of a data collection when it is arranged ascending (lowest value to highest value) or descending (highest value to the lowest value). In other words, the median is the point at which the data array (An array is an arrangement of raw data in ascending or descending order of values) is divided into 2 equal parts. As a result, 50% of the values are below the median, and 50% are above the median.
The Median Calculation Method
When finding the median first, we have to find the median's position using the formula below:
The position of the median = ((n+1)/2)th item
The "n" denotes the overall item count of the data set.
If the total number of items in the dataset is odd, the value of the item at the center position is the median. But suppose the total number of items in the data set is an even figure. In that case, the average between the two numbers in the middle is the median.
Differences Between the Mean and the Median
- The mean, or average, is calculated by summing all the values in a data set and then dividing by the number of observations. It gives us a value that considers each point in the data set. In contrast, the median is the middle value in a data set ordered from lowest to highest and provides a central point that divides the data set in half, but does not take into account the magnitude of all values.
- Both the mean and median can be visually estimated from a graphical representation of data. The mean can be roughly estimated in a symmetric distribution as it should lie at the center, while the median can be determined as the middle value in a box plot, for example.
- Both the mean and median have their uses in further statistical analysis. The mean is particularly useful for data that is normally distributed and does not contain outliers, as it is included in calculations of variance and standard deviation. The median is valuable as a measure of central tendency when the data is skewed or contains outliers, and it is frequently used in non-parametric statistical tests that do not assume a specific data distribution.
When to Use the Mean
The mean is the most suitable measure of central tendency when the data set has a symmetric distribution without outliers, or when outliers have been removed.
When to Use the Median
The mean is not a good representation of the data set when it is skewed by outliers, or when the data set is not symmetrically distributed, or when the data set is skewed. Outliers are data points that are significantly smaller or larger than the other values in the data set. When outliers are present in a data set, the mean or average is greatly affected by these values.
Let's modify the previous example to understand outliers.
Example:
Suppose Jasmine's score for International Studies was 15 instead of 92. What is the average of Jasmine's scores for the previous semester with this new score?
| Subject |
Score |
| Management |
84 |
| Communication |
90 |
| Accounting |
75 |
| Economics |
60 |
| Business Statistics |
85 |
| International studies |
15 |
| Mathematics |
81 |
Solution:
The average score = ΣX / N = (84+90+75+60+85+15+81)/7 = 490/7 = 70
The new average is 70. This is an 11-point decrease from the original average of 81. You can see the effect of the outlier on the average.
In this case, the median of the data is a better measure of central tendency than the mean. To understand this, let's calculate the median for both the original and modified examples.