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# Median

## Middle number of ascending values.

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Median

Mrs. Lee wants to find the median score in her mathematics class. She has organized her students' scores into the following table.

 Student Test Score (%) AS 89 AS 91 BC 87 CS 77 FR 72 JW 59 ML 76 MY 68 ST 91 SR 83 VS 81 ZS 73

How can Mrs. Lee figure out what the median score on the test is?

In this concept, you will learn how to calculate median.

### Calculating Median

Data is a set of numerical or non-numerical information. Data can be analyzed in many different ways. In this concept you will analyze numerical data using the median.

Median is a numerical value that represents the middle term in a data set. When ordered sequentially, the value that is in the middle is the median.

To calculate the median, first order the numbers from smallest to largest. Then, find the number that is in the middle. This number is the median.

Let's look at an example.

Find the median for the set of data: 47, 56, 51, 45, and 41.

First, arrange the numbers from smallest to largest.

41, 45, 47, 51, 56

Next, find the middle number. There are five numbers in this set, so the middle number is the third term, 47.

The answer is the median is 47.

### Examples

#### Example 1

Earlier, you were given a problem about Mrs. Lee and her students' test scores.

Mrs. Lee has given her students a math test and wants to calculate the median score. She has organized her student's data into the following table.

 Student Test Score (%) AS 89 AS 91 BC 87 CS 77 FR 72 JW 59 ML 76 MY 68 ST 91 SR 83 VS 81 ZS 73

First, organize the data from smallest to largest.

59, 68, 72, 73, 76, 77, 81, 83, 87, 89, 91, 91

Next, find the middle term.

There are 12 numbers in this set so the middle number is between the sixth and seventh term, 77 and 81. To find the term that is between 77 and 81, add the numbers together and divide by 2.

\begin{align*}77+81=158\end{align*}

\begin{align*}158\div 2=79\end{align*}

The answer is the median math test score of Mrs. Lee's class is 79%.

#### Example 2

The chart below shows the daily temperature in San Diego for the first seven days in August. Calculate the median temperature.

San Diego Temperatures in August
Date: Temperature:
Sunday 8/1 \begin{align*}88^\circ F\end{align*}
Monday 8/2 \begin{align*}83^\circ F\end{align*}
Tuesday 8/3 \begin{align*}87^\circ F\end{align*}
Wednesday 8/4 \begin{align*}89^\circ F\end{align*}
Thursday 8/5 \begin{align*}82^\circ F\end{align*}
Friday 8/6 \begin{align*}79^\circ F\end{align*}
Saturday 8/7 \begin{align*}87^\circ F\end{align*}

First, arrange the temperatures from smallest to greatest.

79, 82, 83, 87, 87, 88, 89

Next, find the middle number. There are seven numbers in this set, so the middle number is the fourth term, 87.

The answer is the median is \begin{align*}{87}^{o}\end{align*}F.

#### Example 3

Find the median of the following data set.

\begin{align*}12, 14, 15, 16, 18, 20\end{align*}

First, arrange the numbers in order. In this case, they are already ordered from smallest to largest.

Next, find the middle term. There are six numbers in the set, so the middle number is between the third and fourth terms, 15 and 16. To find the term that is between 15 and 16, we add the numbers together and divide by 2.

\begin{align*}15+16=31\end{align*}

\begin{align*}31\div 2=15.5\end{align*}

The answer is the median is 15.5.

#### Example 4

John wants to know his median quiz grade based on the following quiz scores.

78, 90, 83, 88, 67, 90, 84, 69, 56

First, order the numbers from smallest to largest.

56, 67, 69, 78, 83, 84, 88, 90, 90

Next, find the middle number. There are nine numbers in this set, so the middle number is the fifth term, 83.

The answer is the median is 83.

#### Example 5

Find the median of the following data set.

12, 14, 16, 11, 19, 12, 15, 16, 17, 22, 21, 23

First, arrange the numbers from least to greatest.

11, 12, 12, 14, 15, 16, 16, 17, 19, 21, 22, 23

Next, find the middle number. There are 12 numbers in this set, so the middle number is between the sixth and seventh term, 16 and 16. Since these two terms are the same, this number is the median.

The answer is the median is 16.

### Review

Find the median for each set of numbers.

1. 2, 1, 3, 4, 2, 1, 5, 6, 7, 2, 3
2. 11, 12, 17, 18, 21, 12, 13, 13
3. 20, 22, 21, 24, 25, 20, 19
4. 18, 17, 19, 21, 22, 20, 18, 17
5. 19, 29, 39, 49, 59, 69, 79, 89
6. 4, 5, 4, 5, 3, 3, 2, 3, 3, 2
7. 6, 7, 8, 3, 2, 4
8. 11, 10, 9, 13, 14, 16
9. 21, 23, 25, 22, 22, 27
10. 27, 29, 29, 32, 30, 32, 31
11. 34, 35, 34, 37, 38, 39, 39
12. 43, 44, 43, 46, 39, 50
13. 122, 100, 134, 156, 144, 110
14. 224, 222, 220, 222, 224, 224
15. 540, 542, 544, 550, 548, 547

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

cumulative frequency

Cumulative frequency is used to determine the number of observations that lie above (or below) a particular value in a data set.

Data

Data is information that has been collected to represent real life situations, usually in number form.

Mean

The mean of a data set is the average of the data set. The mean is found by calculating the sum of the values in the data set and then dividing by the number of values in the data set.

Median

The median of a data set is the middle value of an organized data set.

normal distributed

If data is normally distributed, the data set creates a symmetric histogram that looks like a bell.

outliers

An outlier is an observation that lies an abnormal distance from other values in a random sample from a population.