Does The Median Represent The Center Of Theã¢â‚¬â€¹ Data?

In this explainer, nosotros volition larn how to find and interpret the median of a data set.

The median is an example of a measure out of middle (or a measure of primal tendency). Oftentimes, we would like to find a unmarried number that can correspond a whole information set or at to the lowest degree give us some information about typical values in the data ready. There are a number of ways to describe typical values. For example, one way to describe a typical value is to see what value is the well-nigh mutual; this is the mode. The mean, which is the average of all the values in a information set up, is another example. We could also describe a typical value past looking at the number in the eye; this is the median.

The mode, mean, and median are all unlike examples of measures of centre. We will merely talk over the median hither.

Definition: The Median

The median of a set of data represents the heart value.

Half of the data values are above the median, and half of the data values are below the median.

Let united states starting time by looking at an example to see how to summate the median when we have an odd number of data values.

Example 1: Finding the Median of a Data Set with an Odd Number of Information Values

Find the median of the values 6, viii, sixteen, 6, and 19.

Answer

The median is the center value. To detect the middle value, we offset put the data values in club from least to greatest:

Now, to find the middle value, nosotros can count in from each side until we reach the middle.

The middle value is 8. Hence, we have found that the median of these five values is 8.

As we saw in the to a higher place case, when there are an odd number of data values, finding the median is piece of cake because there is exactly one middle value.

Nosotros have to think nigh how the situation is different when there are an fifty-fifty number of information values. Let us look at an case that shows what to exercise in this case.

Example ii: Finding the Median of a Data Set up with an Even Number of Information Values

Find the median of the values 13, v, 9, 10, 2, and 15.

Respond

Let us remember that the median of a data set is the value in the center. We will write the numbers in society from to the lowest degree to greatest so we can notice this value:

There are half dozen data values, which is an even number, so there are 2 values in the middle.

To observe the median, we notice the number that is halfway between these middle values. The number halfway between nine and x is

Hence, the median is 9.5 considering half of the data values are higher up information technology and half are beneath it.

Next, we await at what happens when there are an even number of information values with a repeated value in the middle.

Example three: Finding the Median of a Data Prepare with a Repeated Centre Value

What is the median of the numbers 11, eleven, 8, eight, ix, and ix?

Answer

Recall that the median represents the heart of the information.

We start by writing the numbers in order and finding the middle value (or values).

Normally, when there are two heart values, we take to summate the number halfway between them. However, since both middle numbers are the same here, this value is the median.

Hence, the median is nine.

Now that we have seen some examples of how to find the median of a data set, allow us summarize what we have learned.

How To: Finding the Median of a Data Set

Write the information in order from least to greatest.
Find the middle value (or values).
If there are an odd number of data values, there will be exactly one value in the middle.
If there are an even number of information values, there will be two values in the middle.
Discover the median.
If at that place are an odd number of information values, the median is the heart value.
If there are an fifty-fifty number of data values, the median is halfway between the 2 middle values.

In the in a higher place examples, it was relatively uncomplicated to order the data values from least to greatest and then piece of work out the center value (or values). When we have larger sets of data, however, we need to make certain nosotros count the number of values given in the question so that nosotros exercise not omit any by blow when reordering them. Also, if the data values include decimal numbers, we demand to brand sure that we copy them downward accurately, reorder them correctly, and carry out whatsoever calculations carefully. Here is an example of this blazon.

Example four: Finding the Median of a Larger Data Set with an Even Number of Values

Calculate the median of the values two.9, v.4, 5.one, iii.7, iii.4, 1.9, half dozen.one, eight.one, 12.4, 2.6, eight.8, and 5.5.

Answer

Think that the median of a data set is the middle value, which we tin observe by ordering the values and finding the single value (or pair of values) in the center.

In this question, we have 12 data values. Since 12 is an even number, when nosotros social club the values from least to greatest, there will be two values in the centre. The median will exist the value halfway between these two.

Writing the data values from least to greatest, we accept ii middle values, as shown below.

As the two heart values are v.1 and 5.4, then the number halfway between them is

Therefore, five.25 is the median of this information set.

The fact that the median is the middle value tin can help us solve problems where in that location is a missing data value, equally shown in the adjacent example.

Example 5: Finding an Unknown Value in a Data Set with a Known Median

Farida has the following information: ten, 8, 7, 9, .

If the median is 8, what number could be?

8.five
7
9
x
nine.5

Respond

Recall that the median represents the eye of the data. Let united states of america start past ordering the information values from least to greatest and seeing what clues nosotros tin effigy out about the number .

Nosotros are told that the median is 8, and we know that there must be the same number of information values above the median as below it. As there are already two values above the median, the number tin exist at nearly the median value.

This means that can be any number that is 8 or below. However, out of the available choices, we see that none of viii.5, ix, 9.5, or 10 volition work.

Therefore, we must have that .

Note that, in the above instance, the fact that there were already two values above the median number of 8 did not mean that we could immediately assume the missing value must be less than 8. We could only assume it was at near viii. This is considering a information set such every bit would all the same have a median of 8, even though only i of the information values is strictly less than the median value. In our example, the missing value turned out to be less than 8 because 7 was the merely one of the five available reply options that satisfied the status of existence at most 8.

When working out the median for data sets involving a mixture of positive and negative numbers, information technology is especially important to brand sure the numbers are ordered correctly, without dropping any minus signs. Let us await at an case.

Example 6: Finding the Median of a Information Set Involving Negative Numbers

Nader's game scores were , ii, 2, , , , and viii. Decide the median.

Answer

Nosotros have been asked to notice the median of some data, which we can recall is the value that lies in the heart of the other values when they are ordered.

Here, the information values are game scores. The number of data values is seven, which is an odd number. Therefore, when we order the values from least to greatest, in that location will exist a single value in the middle; this is the median.

Writing the data values from least to greatest, not forgetting any minus signs, we get one middle value, as shown below.

We conclude that the median of Nader'southward game scores was .

In our concluding example, we can apply the ideas explored here to calculate the medians of two data sets listed in a table and then find the divergence between the two medians.

Example 7: Calculating the Median for Two Data Sets and Finding the Difference between the Two Values

The table records the heights, in inches, of a group of fifth graders and a group of sixth graders. What is the difference between the medians of the heights of both groups?

Fifth Grade
Sixth Class

Answer

Recall that the median represents the heart of the data.

In this question, nosotros have ii data sets made up of the heights, in inches, of two groups of students. Each data set up has 7 values, which is an odd number. This means that in both cases, the median volition be the single value in the heart that we obtain when we order the data values from least to greatest.

Writing the 5th course heights from least to greatest, we get the following list.

Therefore, the median pinnacle of the fifth graders is 61 inches.

Similarly, writing the sixth graders' heights from to the lowest degree to greatest, we get another list.

From this, we see that the median tiptop of the sixth graders is lx inches.

Finally, we work out the difference between the medians by subtracting the smaller number from the larger 1, to get . Thus, nosotros have found that the difference between the medians of the heights of the groups is 1 inch.

Let us finish by recapping some key concepts from this explainer.

Fundamental Points

The median of a set of data represents the middle value.
If there are an odd number of information values, there will be exactly one value in the middle; this is the median.
If in that location are an even number of data values, at that place will be two values in the heart; the value halfway between the 2 is the median.
E'er make sure that you lot count the number of data values given in the question so that none are omitted accidentally when we reorder them from least to greatest.
If the data values include decimal numbers or negative numbers, make sure you re-create them down accurately and reorder them advisedly.
When given the median of a data set, we can sometimes piece of work backward to notice an unknown data value.