4.5: Median
- Page ID
- 5719
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let's Think About It
James Wragg - www.flickr.com/photos/jameswragg/4688532009/in/photolist-89iXdM-ao9GDP-9fzpg2-h2vxw8-9y6LSZ-e1VCAb-cHLKcy-fd7vPF-QJ4NT-ab1EXN-c128kq-5tCH1E-98S9zG-eLr9pc-9DYaVa-cWxtYy-9y9KuU-nwXTDb-8Jk2od-p4hE6h-fd7sPM-9F5Ae8-nxW7tt-h2uKHA-e4NP8x-oCfxLz-pqsmam-h2tLqE-8GGFRw-7qonhm-dmKd9J-e91zeG-nwY1gk-rxbJP-9vT4n5-7vCfq5-btCJDZ-4ZQMbc-7Fihq9-73Pjgh-dmK2CR-aE44a1-2Y3Xp-donWVX-dnBdhf-6Vvw1D-aNR4z-vPoiV-dGJvYU-dGJxn5
Sara's cat has a new litter of kittens each year. She has been keeping track of how many kittens are in each litter. Sara wants to know what the typical amount of kittens is in each litter. To do this, she could find the middle amount of kittens that have been born. Here is Sara's data. Each number represents an amount of kittens that were born in one year.
7, 8, 9, 12, 14, 10, 15, 14
How can Sara find that middle number in her data set?
In this concept, you will learn how to find the median of a set of data.
Guidance
The median of a set of data is the middle score or number of the data. Median can be useful in examples like finding the median amount of money or median time a runner takes to run a race.
Here is a set of data. To find the median there are a few steps.
2, 5, 6, 2, 8, 11, 13, 14, 15, 21, 22, 25, 27
First, write the numbers in order from the least to greatest. Be sure to include repeated numbers in the list.
2, 2, 5, 6, 8, 11, 13, 14, 15, 21, 22, 25, 27
Next, find the middle number of the set of data.
In this set, there is an odd number of values in the set. There are thirteen numbers in the set. Count 6 on one side of the median and 6 on the other side of the median.
The answer is 13.
This set of data was easy to work with because there was an odd number of values in the set. It does not always work out that way, though. Sometimes, there is an even number of items in the data set.
4, 5, 12, 14, 16, 18
Here, there are six values in the data set. They are already written in order from least to greatest. This data set has two values in the middle because there are six values.
4, 5, 12, 14, 16, 18
The two middle values are 12 and 14. To find the median, add the two middle values together and divide by 2. This is basically taking an average of the two middle values.
12+14=26
26÷2=13
The median score is 13.
Guided Practice
Jess has planted a garden. His big crop has been eggplant. Jess harvested the following numbers of eggplant over five days. What is the median number of eggplant harvested?
12, 9, 15, 6, 9
First, write the numbers in order from least to greatest.
6, 9, 9, 12, 15
Next, count two numbers on each side of the middle of the data set to narrow down to the median number.
Then, find the middle score.
The answer is 9 eggplants in the median number harvested.
Examples
Find the median of each data set.
Example 1
11, 5, 8, 6, 15
First, order the numbers from least to greatest.
5, 6, 8, 11, 15
Next, count the total number of items in the data set to determine how many to count in from each side.
Then, count in until the median is reached.
The answer is 8.
Example 2
4, 1, 6, 9, 2, 11
First, order the numbers from least to greatest.
1, 2, 4, 6, 9, 11
Next, count the total number of items in the data set to determine how many to count in. In this case, there are 6 numbers in the data set, which means an even number of items.
Then, since there are an even number of values in the data set, an average needs to be taken of the two middle values.
4+6=10
10÷2=5
The answer is 5.
Example 3
63, 23, 78, 34, 56, 89
First, order the numbers from least to greatest.
23, 34, 56, 63, 78, 89
Next, count the total number of items in the data set. There is an even number of items in the data set.
Then, since there are 6 values in the set, take an average of the middle two values to determine the median.
56+63=119
119÷2=59.5
The answer is 59.5
Follow Up
Laitche - https://en.Wikipedia.org/wiki/Kitten#/media/File:Laitche-P013.jpg
Remember Sara and her kittens? She wants to find the middle value of a data set, which is known as finding the median. Here is Sara's data that shows how many kittens were born in each litter:
7, 8, 9, 12, 14, 10, 15, 14
First, Sara needs to put her data in order from least to greatest.
7, 8, 9, 10, 12, 14, 14, 15
Next, Sara needs to count the number of items in her data set in order to continue on following the correct steps. There are 8 values in her data set, an even number.
Then, Sara has to find the two middle values, since there is an even number, and then find the average of those two middle values.
10+1222÷2=22=11
The answer is 11. The median number of kittens born each year is 11 per litter.
Video Review
Explore More
Find the median for each pair of numbers.
1. 16 and 19
2. 4 and 5
3. 22 and 29
4. 27 and 32
5. 18 and 24
Find the median for each set of numbers.
6. 4, 5, 4, 5, 3, 3
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
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 2.19.
Vocabulary
| Term | Definition |
|---|---|
| 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. |
Additional Resources
Video: Mean, Median, and Mode
Activities: Median Discussion Questions
Study Aids: Describing Data
Lesson Plans: The Median Lesson Plan
Practice: Median
Real World: Medians