Average Calculator (Mean, Median, Mode, Range)

The Mean, Median, and Mode are the Main Measures of Central Tendency.

The mean represents the arithmetic mean, the median represents the middle value when the numbers are arranged in order, and the mode identifies the number that occurs most frequently. If the distribution is perfectly balanced, the mean and median will be equal.

When the distribution is perfectly symmetrical and has only one mode, the mean, median, and mode all share the same value.

Mean:

The word “mean,” which shares spellings with many other English words, also has various interpretations in mathematics and statistics. Its precise meaning depends on the context in which it is used. In its most basic mathematical sense, mean refers to the arithmetic mean, also called the average or expected value of a set of numbers. In this case, the mean represents the central value of a data set, which is calculated by adding all the numbers and dividing by the total count of the numbers.

Median:

In statistics, the median represents the value that divides a data sample, population, or probability distribution into two equal parts. To determine the median, you identify the number that is in the middle of a ranked data set. The order of the numbers is important when calculating the median. Typically, the numbers are arranged in ascending order, although descending order gives the same median. If a data set has an odd number of values, the median is simply the central number in the list.

Mode:

In statistics, the mode is the number that appears most frequently in a data set. Some data sets may have more than one mode, in which case they are called multimodal.

Range:

The range of a data set measures the difference between its largest and smallest values. While the term range may have other interpretations in mathematics and statistics, this is its basic definition and the definition applied by the calculator mentioned.

How to Find the Mean – Average Formula

When people refer to the mean, they usually mean the arithmetic mean, which is the average of all the numbers in a dataset. If you’re wondering, “How do you find the mean?” here’s a clear explanation:

The formula for the arithmetic mean is:

μ = ∑X / N

where:

• μ is the population mean (for a sample, you can use M instead, but the calculation remains the same);
• ∑X is the sum of all the numbers; and
• N is the total number of numbers..

For example, to find the mean of 2, 5, 6, 8, 9:

μ = (2 + 5 + 6 + 8 + 9) / 5

μ = 30 / 5

μ = 6

How to Find the Median

To find the median, consider the numbers 2, 7, 4, 8, 6, 9:

Sort the numbers from smallest to largest

The order becomes:

2, 4, 6, 7, 8, 9

Sorting from smallest to largest works perfectly.

Identify the center of the numbers

Here the middle two numbers are 6 and 7, but for larger sets, calculating the center is helpful.

Use this formula to find the median:

Median = (N + 1) / 2

Since there are six numbers:

Median = (6 + 1) / 2 = 3.5

This means that the median is the average of the 3rd and 4th numbers.

Determine the median

Counting from the list, the 3rd and 4th numbers are 6 and 7:

2, 4, 6, 7, 8, 9

The median is (6 + 7) / 2 = 6.5.

How to Find the Mode

Let’s find the mode using three approaches with the numbers: 2, 7, 6, 4, 7, 7, 6, 8, 7, 4, 6

Method 1: Tick marks on the number line

Draw a number line and place tick marks for each number.

The largest column number is the mode, which here is 7.

Method 2: Arrange the numbers from smallest to largest

Sorting helps you quickly identify the mode:

2, 4, 4, 6, 7, 7, 8

How to Calculate the Range and Median Range – Range Formula

The range is the simplest measure of spread in a dataset. Find it by subtracting the smallest value from the largest using this formula:

Range = Maximum – Minimum

The median range is another measure of central tendency, like the mean, median, and mode. It lies exactly halfway between the smallest and largest numbers:

Median range = (maximum + minimum) / 2

For the numbers 2, 7, 6, 4, 7, 7, 6, 8, 7, 4, 6, the smallest value is 2 and the largest value is 8.

Therefore, range = 8 – 2 = 6, and middle range = (8 + 2) / 2 = 5.

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