5.4: Median
Here are the temperatures you recorded for the last fifteen days in degrees Fahrenheit: 25, 55, 65, 60, 10, 52, 61, 58, 53, 67, 61, 63, 64, 59, 47. What measurement would you use to describe a representative temperature over those fifteen days?
Watch This
First watch this video to learn about the median.
CK-12 Foundation: Chapter5MedianA
Then watch this video to see some examples.
CK-12 Foundation: Chapter5MedianB
Watch this video for more help.
This video from Khan Academy shows how to calculate the mean, median, and mode of a data set.
Khan Academy Average or Central Tendency: Arithmetic Mean, Median, and Mode
Guidance
Before class begins, bring out the blocks that you and your classmates chose from the pail for the concept on mean. In addition, have the grid paper on display where each student in your class posted his or her number of blocks.
To begin the class, refer to the comments on the measures of central tendency that were recorded from the earlier concept, when the brainstorming session occurred. Highlight the comments that were made with regard to the median of a set of data and discuss this measure of central tendency with your classmates. Once the discussion has been completed, choose a handful of blocks like you did before, with your classmates doing the same.
Next, form a line with your classmates as a representation of the data from smallest to largest. In other words, those with the largest number of blocks should be at one end of the line, and those with the fewest number of blocks should be at the other end of the line. Have those at the ends of the line move out from the line 2 at a time, with 1 from each end leaving the line at the same time. This movement from the ends should enforce the concept of what is meant by center, and as you and your classmates move away, you will see that the last 1 or 2 people remaining in the line actually represent the center of the data.
A similar activity can be done by using the grid paper chart. Instead of you and your classmates moving from a line, you could simply remove your post-it notes the same way.
Another simple activity to reenforce the concept of center is to place 5 desks in a single row and to have one of your classmates sit in the middle desk.
Your classmate should be sitting in desk number 3. The other students in your class will quickly notice that there are 2 desks in front of your classmate and 2 desks behind your classmate. Therefore, your classmate is sitting in the desk in the middle position, which is the median of the desks.
From the discussion, the activity with the blocks, and the activity with the desks, you and your classmates should have an understanding of the meaning of the median with respect to a set of data. Here is another example. The test scores for 7 students were 25, 55, 58, 64, 66, 68 and 70. The mean mark is 48.6, which is lower than all but 1 of the student’s marks. The one very low mark of 25 has caused the mean to be skewed. A better measure of the average performance of the 7 students would be the middle mark of 64. The median is the number in the middle position once the data has been organized. Organized data is simply the numbers arranged from smallest to largest or from largest to smallest. 64 is the only number for which there are as many values above it as below it in the set of organized data, so it is the median. The median for an odd number of data values is the value that divides the data into 2 halves. If \begin{align*}n\end{align*} represents the number of data values and \begin{align*}n\end{align*} is an odd number, then the median will be found in the \begin{align*}\frac{n+1}{2}\end{align*} position.
The median is often preferable to the mean as a summary statistic, because the mean is affected by extreme values, or outliers. For example, if you were employed by a company that paid all of its employees a salary between $60,000 and $70,000, you could probably estimate the mean salary to be about $65,000. However, if you had to add in the $150,000 salary of the CEO when calculating the mean, then the value of the mean would increase greatly. It would, in fact, be the mean of the employees' salaries, but it probably would not be a good measure of the central tendency of the salaries. It would be much better to use the median as a summary statistic in this case.
Example A
Find the median of the following data:
a) 12, 2, 16, 8, 14, 10, 6
b) 7, 9, 3, 4, 11, 1, 8, 6, 1, 4
a) The first step is to organize the data, or arrange the numbers from smallest to largest.
\begin{align*}12, 2, 16, 8, 14, 10, 6 \qquad \rightarrow \qquad 2, 6, 8, 10, 12, 14, 16\end{align*}
The number of data values is 7, which is an odd number. Therefore, the median will be found in the \begin{align*}\frac{n+1}{2}\end{align*} position.
\begin{align*}\frac{n+1}{2}=\frac{7+1}{2}=\frac{8}{2}=4\end{align*}
In this case, the median is the value that is found in the \begin{align*}4^{\text{th}}\end{align*} position of the organized data.
\begin{align*}2, 6, 8, \boxed{10}, 12, 14, 16\end{align*}
This means that the median is 10.
b) The first step is to organize the data, or arrange the numbers from smallest to largest.
\begin{align*}7, 9, 3, 4, 11, 1, 8, 6, 1, 4 \qquad \rightarrow \qquad 1, 1, 3, 4, 4, 6, 7, 8, 9, 11\end{align*}
The number of data values is 10, which is an even number. Therefore, the median will be the mean of the number found before the \begin{align*}\frac{n+1}{2}\end{align*} position and the number found after the \begin{align*}\frac{n+1}{2}\end{align*} position.
\begin{align*}\frac{n+1}{2}=\frac{10+1}{2}=\frac{11}{2}=5.5\end{align*}
The number found before the 5.5 position is 4, and the number found after the 5.5 position is 6.
\begin{align*}1, 1, 3, 4, \boxed{4, 6}, 7, 8, 9, 11\end{align*}
This means that the median is \begin{align*}\frac{4+6}{2}=\frac{10}{2}=5\end{align*}.
Example B
The amount of money spent by each of 15 high school girls for a prom dress is shown below:
\begin{align*}& \$250 && \$175 && \$325 && \$195 && \$450 && \$300 && \$275 && \$350 && \$425\\ & \$150 && \$375 && \$300 && \$400 && \$225 && \$360\end{align*}
What is the median price spent on a prom dress?
\begin{align*}& \$150 && \$175 && \$195 && \$225 && \$250 && \$275 && \$300 && \boxed{\$300} && \$325 && \$350 && \$360 && \$375\\ & \$400 && \$425 && \$450\end{align*}
The prices have been organized from least to greatest, and the number of prices is an odd number. Therefore, the median will be in the \begin{align*}\frac{n+1}{2}\end{align*} position: \begin{align*}\frac{n+1}{2}=\frac{15+1}{2}=\frac{16}{2}=8\end{align*}.
The median price is $300, which is the \begin{align*}8^{\text{th}}\end{align*} position.
Example C
The following table shows the number of goals scored by Ashton during each of 26 hockey games. What is the median number of goals scored by Ashton during a game?
When data is entered into a frequency table, a column that displays the cumulative frequency is often included. This column is simply the sum of the frequencies up to and including that frequency. The median can be determined by using the information that is presented in the cumulative frequency column.
In this example, there are 26 data values, which is an even number. The middle position is \begin{align*}\frac{n+1}{2}=\frac{26+1}{2}=\frac{27}{2}=13.5\end{align*}, and the median is the sum of the numbers above and below position 13.5 divided by 2. According to the table, the numbers in the \begin{align*}13^{\text{th}}\end{align*} and \begin{align*}14^{\text{th}}\end{align*} positions are 2’s. Therefore, the median is \begin{align*}\frac{2+2}{2}=\frac{4}{2}=2\end{align*} goals.
Points to Consider
- The median of a set of data values cuts the data in half. Is only the median of an entire set of data a useful value?
- Is the median of a set of data useful in any other aspect of statistics?
- Other than as a numerical value, can the median be used to represent data in any other way?
Guided Practice
The students from a local high school volunteered to clean up the playground as an act of community service. The numbers of pop cans collected by 20 of the students are shown in the following table:
\begin{align*}& 16 \quad 22 \quad 10 \quad 8 \quad \ 14\\ & 12 \quad 36 \quad 18 \quad 12 \quad 10\\ & 34 \quad 26 \quad 44 \quad 6 \quad \ 20\\ & 31 \quad 25 \quad 15 \quad 9 \quad \ 13\end{align*}
What is the median number of pop cans collected by a student?
Answer:
\begin{align*}& 6 \quad \ \ 8 \quad \ 9 \quad \ 10 \quad 10\\ & 12 \quad 12 \quad 13 \quad 14 \quad 15\\ & 16 \quad 18 \quad 20 \quad 22 \quad 25\\ & 26 \quad 31 \quad 34 \quad 36 \quad 44\end{align*}
There is an even number of data values in the table, so the median will be the mean of the number before the \begin{align*}\frac{n+1}{2}\end{align*} position and the number after the \begin{align*}\frac{n+1}{2}\end{align*} position: \begin{align*}\frac{n+1}{2}=\frac{20+1}{2}=\frac{21}{2}=10.5\end{align*}.
The number before the 10.5 position is 15, and the number after the 10.5 position is 16. Therefore, the median number of pop cans collected by a student is \begin{align*}\frac{15+16}{2}=\frac{31}{2}=15.5\end{align*}.
Interactive Practice
Practice
- What name is given to a value in a data set that is much lower or much higher than the other values?
- A sample
- An outlier
- A population
- A tally
- What is the median of the following numbers? \begin{align*}10, 39, 71, 42, 39, 76, 38, 25\end{align*}
- 42.5
- 39
- 42
- 35.5
- The front row in a movie theatre has 23 seats. If you were asked to sit in the seat that occupied the median position, in what number seat would you have to sit?
- 1
- 11
- 23
- 12
- What is the median mark achieved by a student who recorded the following marks on 10 math quizzes? \begin{align*}68, 55, 70, 62, 71, 58, 81, 82, 63, 79\end{align*}
- 68
- 71
- 69
- 79
- A set of 4 numbers that begins with the number 32 is arranged from smallest to largest. If the median is 35, which of these could possibly be the set of numbers?
- 32, 34, 36, 38
- 32, 35, 38, 41
- 32, 33, 34, 35
- 32, 36, 40, 44
- The number of chocolate bars sold by each of 30 students is as follows: \begin{align*}32, 6, 21, 10, 8, 11, 12, 36, 17, 16, 15, 18, 40, 24, 21, 23, 24, 24, 29, 16, 32, 31, 10, 30, 35, 32, 18, 39, 12, 20\end{align*} What is the median number of chocolate bars sold by the 30 students?
- 18
- 21
- 24
- 32
- The following table lists the retail price and the dealer’s costs for 10 cars at a local car lot this past year: Deals on Wheels
Car Model | Retail Price | Dealer’s Cost |
---|---|---|
Nissan Sentra | $24,500 | $18,750 |
Ford Fusion | $26,450 | $21,300 |
Hyundai Elantra | $22,660 | $19,900 |
Chevrolet Malibu | $25,200 | $22,100 |
Pontiac Sunfire | $16,725 | $14,225 |
Mazda 5 | $27,600 | $22,150 |
Toyota Corolla | $14,280 | $13,000 |
Honda Accord | $28,500 | $25,370 |
Volkswagen Jetta | $29,700 | $27,350 |
Subaru Outback | $32,450 | $28,775 |
(a) Calculate the median for the data on the retail prices for the above cars.
(b) Calculate the median for the data on the dealer’s costs for the above cars.
- Due to high winds, a small island in the Atlantic suffers frequent power outages. The following numbers represent the number of outages each month during the past year: \begin{align*}4 \quad 5 \quad 3 \quad 4 \quad 2 \quad 1 \quad 0 \quad 3 \quad 2 \quad 7 \quad 2 \quad 3\end{align*} What is the median number of monthly power outages?
- The Canadian Coast Guard has provided all of its auxiliary members with a list of 14 safety items (flares, fire extinguishers, life jackets, fire buckets, etc.) that must be aboard each boat at all times. During a recent check of 15 boats, the number of safety items that were aboard each boat was recorded as follows: \begin{align*}7 \quad 14 \quad 10 \quad 5 \quad 11 \quad 2 \quad 8 \quad 6 \quad 9 \quad 7 \quad 13 \quad 4 \quad 12 \quad 8 \quad 3\end{align*} What is the median number of safety items aboard the boats that were checked?
- A teacher’s assistant who has been substituting has been recording her biweekly wages for the past 13 pay periods. Her biweekly wages during this time were the following: \begin{align*}& \$700 \qquad \$550 \qquad \$760 \qquad \$670 \qquad \$500 \qquad \$925 \qquad \$600\\ & \$480 \qquad \$390 \qquad \$800 \qquad \$850 \qquad \$365 \qquad \$525\end{align*} What is her median biweekly wage?
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Color | Highlighted Text | Notes | |
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Term | Definition |
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cumulative frequency | Cumulative frequency is used to determine the number of observations that lie above (or below) a particular value in a 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. |