# Median – Tutorial, Calculations & Examples

02.01.23 Time to read: 4min
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The median is an essential concept in statistics and represents a type of measure of central tendency of a set of numerical data. Unlike the mean or mode averages, this type of average is calculated by working out where the middle of a set of data is, also known as the midpoint. This article provides a thorough understanding of the median and its central tendency of skewed distributions by demonstrating its calculations and significance in statistical analysis.

## Median – In a Nutshell

• The median is a statistical measure that represents the exact middle or midpoint of a set of data. It’s part of the measures of central tendency, along with the mean and mode.
• Calculating the median involves arranging the data in ascending order and identifying the middle value. If there’s an odd number of data points, the median is the exact middle value. If there’s an even number, it’s the average of the two middle values.
• The median can provide different insights than the mean or mode, particularly in data sets with extreme values, because these values do not affect it. This makes it a useful measure in understanding the central tendency of skewed distributions.
• For normally distributed data, the median, mean, and mode coincide due to the symmetric nature of the distribution. This isn’t the case for skewed distributions.

## Definition: Median

The median is the middle number in a set of data.

Example

If you had a set of data that, when arranged in ascending order was

• “3, 5, 6, 9, 12”

then it would be six, which was the average. For this reason, median values should not be confused with mean and mode averages, or standard deviation.

In the aforementioned example, the median is six while the mean average would be seven, which is relatively close. However, in another data set – say 3, 5, 6, 91, 120 – it would still be six while the mean average would be 45.

Example

 Type of Average Mean Average Mode Average Median Average Meaning The sum of a set divided by the number of terms in that set The most frequently occurring number in a set The midpoint in a set

Note: The median averages can yield very different results.

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## Median calculation

Whether a set-up data is small or large, the calculation method is the same. You still just have to arrange all the data in order and pick the middle value. That’s straightforward for odd-numbered data sets.

However, if there is an even number of values in a given set of statistical data, then there will be no single value in the middle.

Example

If you have 100 numbers in your data set, then you might imagine that both the 50th and 51st values could be used as the midpoint value.

Therefore, when even-numbered sets of data are used, just add them both together and divide by two.

In other words, the mean average of both middle values is calculated to produce the true median in even-numbered data sets.

Let’s imagine a set of data based on UK shoe sizes.

From a survey of recent shoe sales, it is established that on a given day, pairs of shoes in sizes 3, 4 ½, 9, 9 ½, 7, 12, 5, 6, and 5 were sold. In other words, there were nine pairs of shoes sole with the most frequently occurring being size five.

Example

1. Rearrange the data in ascending order, that is to say, 3, 4 ½, 5, 5, 6, 7, 9, 9 ½, and 12.
2. Since the number five is midway between one and nine, it will the fifth number in the ordered set, which is the median.
3. Count along the ordered dataset from left to right until you reach the fifth number. In this example, it is six which is the midpoint value.

In another example, using an even number of values, we’ll imagine the finishing times of ten runners who have completed a circuit of a track.

Example

Their times are 1 min 9s, 56s, 1 min 10s, 48s, 1 min 19s, 1 min 2s, 59s, 1 min 11s, 1 min 15s and 1 min dead.

To work out the midpoint average:

1. Rearrange the data in ascending order simplified into seconds only, that is to say, 48, 56, 59, 60, 62, 69, 70, 71, 75 and 79.
2. Since the fifth and sixth numbers in the ordered list could both be the midpoints, find them both. In this example, 62 and 69 are the correct ones to use.
3. Add them together and divide them by two. In this example, that would be (62+69) ÷ 2 = 65.5. Therefore, 65.5 seconds is the midpoint value.

## The median in a normal distribution

Normally, distributed sets of data can be represented easily on a graph. When a normal distribution makes a bell curve on a graph, the midpoint between the two tips of the bell will be the median value.

Because bell curves formed by normal distributions are symmetrical, the midpoint on the x-axis of such a graph will also coincide with the highest point on the y-axis. As such, when data sets are normally distributed, the midpoint average value will also be the most frequently occurring. In other words, the median, mean, and mode averages will all be the same.

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## FAQs

#### What is the difference between the median and the mean?

Mean averages are calculated by adding up all the values in a set and dividing that sum by how many values there are, whereas the median is a straightforward midpoint between the highest and lowest values.

#### Why is using a midpoint average a good idea?

Median averages that rely on midpoints are good for establishing averages that do not take account of extremes at either end of the scale. In short, they tend to ignore outlier statistics.

#### What is the median formula?

With an odd set of data, the midpoint formula to use is:

,

while the formula

should be applied to even sets of data.

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