1. Median and Mode
Lesson 5.1.1

2. Median and Mode 5.1.1 California Standard: What it means for you:
Lesson
5.1.1
Median and Mode
California Standard:
Statistics, Data Analysis, and Probability 1.1
Compute the range, mean, median, and mode of data sets.
What it means for you:
You’ll learn to find the median and mode for sets of data.
Key words:
central tendency
data
median
mode
values

3. Lesson
5.1.1
Median and Mode
The median and mode are both intended to “summarize” a whole data set in a single number.
They should show some kind of “most usual” or “middle” value of a set of data.
But although they’re similar in some ways, they’re worked out very differently. This Lesson is about how to find them.

4. Median and Mode 5.1.1 Data Sets Can Contain All Sorts of Values
Lesson
5.1.1
Median and Mode
Data Sets Can Contain All Sorts of Values
Data sets often contain numerical values.
For example, the data set below represents the number of hours eight adults said they slept last night.
Braces “{“ and “}” are used to show that values are grouped together in a set.
{12, 8, 7, 8, 8.5, 8, 9, 6}
Data sets can contain other types of information too.
For example, the data set below represents the hair color of five students.
{blonde, brunette, red, blonde, black}
If there are two items the same, they’re listed twice.

5. Lesson
5.1.1
Median and Mode
Data sets can contain huge amounts of data, and it’s very likely that most people won’t be interested in reading every single value.
So, often a value that represents a typical value for the set is used. These typical values are often referred to as measures of central tendency.

6. Median and Mode 5.1.1 The Median Is a Measure of Central Tendency
Lesson
5.1.1
Median and Mode
The Median Is a Measure of Central Tendency
The Median
The median of a data set is a value that divides the set into two equal groups — one group containing values bigger than the median, the other containing smaller values.
So the median is the middle value when a set of values is put in order.
If you have an odd number of values, the median is fairly easy to find.

7. Median and Mode 5.1.1 Find the median of the following data set:
Lesson
5.1.1
Median and Mode
Example 1
Find the median of the following data set:
{3, 2, 6, 8, 2, 10, 6, 4, 9}
Solution
First, arrange the values in order: {2, 2, 3, 4, 6, 6, 8, 9, 10}
There are nine values, so the median is the fifth value.
There are four values less than the median...
... and four values greater than the median.
{2, 2, 3, 4, 6, 6, 8, 9, 10}
median
The median of the data set is 6.
Solution follows…

8. Lesson
5.1.1
Median and Mode
If there’s an even number of values, then finding the median is slightly trickier because there are two middle numbers.
Here, you find the value exactly midway between the two middle numbers.

9. Lesson
5.1.1
Median and Mode
Example 2
Find the median of this data set: {4.6, 8.9, 9, 10, 10, 14.7}
Solution
The values in this data set are already in order from least to greatest.
There are six values in the set — the median lies midway between the third and fourth values.
Median = 9.5
There are three values below the median...
... and three values above the median.
{4.6, 8.9, 9, 10, 10, 14.7}
The third and fourth values are 9 and 10 — so the median is 9.5.
Solution follows…

10. Median and Mode 5.1.1 Guided Practice
Lesson
5.1.1
Median and Mode
Guided Practice
Find the median of each of the following data sets.
1. {12, 8, 10, 19, 21, 7, 14} 2. {$101, $201, $150, $198, $300} 3. {5, 8, 3, 6, 12, 9, 5, 5, 4, 11} 4. {–6, –3, 7, 4, –2, –2, 5, 2} 5. {–2.1, 5.7, 8.1, –10.2, –100, } 12
$198
5.5
1.8
Solution follows…

11. Median and Mode 5.1.1 The Mode Is Another Measure of Central Tendency
Lesson
5.1.1
Median and Mode
The Mode Is Another Measure of Central Tendency
The Mode
The mode of a data set is the value that occurs most often.
To find the mode of a data set, look for the value that’s listed more than any other value.

12. Median and Mode 5.1.1 Find the mode of the data set:
Lesson
5.1.1
Median and Mode
Example 3
Find the mode of the data set:
{17, 2, 6, 8, 2, 10, 4, 35, 10, 7, 2}
Solution
The number 2 occurs three times — this is more than any other value in the set.
{17, 2, 6, 8, 2, 10, 4, 35, 10, 7, 2}
So the mode is 2.
Solution follows…

13. Median and Mode 5.1.1 Some Data Sets Have No Modes — Others Have Many
Lesson
5.1.1
Median and Mode
Some Data Sets Have No Modes — Others Have Many
Data sets don’t always have one mode — as the next two examples show.

14. Lesson
5.1.1
Median and Mode
Example 4
Find the mode of this set of data: {brown, blue, green, blue, yellow, brown, orange, white}
Solution
Blue and brown both appear twice. No other color appears more often.
So the data set has two modes, blue and brown.
Solution follows…

15. Lesson
5.1.1
Median and Mode
Example 5
Find the mode of this set of data: {3, 5, 19, 5, 3, 19}
Solution
Each number occurs twice — no value occurs more often than the others.
So this data set has no modes.
Solution follows…

16. Median and Mode 5.1.1 Guided Practice
Lesson
5.1.1
Median and Mode
Guided Practice
Give the mode(s) of the following data sets. 6. {$12, $8, $7.50, $7.50, $10, $8, $8, $9.50} 7. { , , ,1, }
8. {0, 1, –3, 5, 1, 0, –3, –3, 0} $8 2 5 1 2 1 2 3 4 1 2
–3 and 0
Solution follows…

17. Median and Mode 5.1.1 Independent Practice
Lesson
5.1.1
Median and Mode
Independent Practice
Give the median and mode(s) of the data sets in Exercises 1–4.
1. {42, 56, 73, 64, 42}
2. {2, 5, 7, 3, 8, 10, 14}
3. {$16, $28, $20, $15}
4. {0.1, 0.4, 0.7, 0.4, 0.5, 0.7}
Median = 56, mode = 42
Median = 7, no mode
Median = $18, no mode
Median = 0.45, modes = 0.4 and 0.7
5. Write a set of data for which the mode and the median are the same.
There are many answers, for example {3, 3, 3, 3, 3}.
Solution follows…

18. Median and Mode 5.1.1 Independent Practice
Lesson
5.1.1
Median and Mode
Independent Practice
Exercises 6–7 are about Rick’s survey of car colors.
6. Rick listed the colors of the 25 cars in a parking lot. The mode for his list is blue. What does this tell you?
There were more blue cars than cars of any other color.
7. There are only 5 car colors on Rick’s list: white, red, black, blue, and green. The mode is blue. What can you say about the possible minimum and maximum number of blue cars? Explain your answer.
The maximum number of blue cars is 21 (so that the other colors appear at least once). The minimum number of blue cars is 6 (since if there were any fewer, blue couldn’t be the mode).
Solution follows…

19. Median and Mode 5.1.1 Independent Practice
Lesson
5.1.1
Median and Mode
Independent Practice
In Exercises 8–11, find a number for each blank so the median is 12.
8. {42, 2, 5, 7, 36, __}
9. {4, 12, 18, __} 10. {2, 5, 14, 12, __}
11. {6, 6, 7, 9, 11, 17, 17, 20, 23, __} 17 12
Any number greater than or equal to 12 13
Solution follows…

20. Median and Mode 5.1.1 Independent Practice
Lesson
5.1.1
Median and Mode
Independent Practice
Decide whether each statement is true or false. Explain your answers.
12. The median of a data set always equals one of the data values.
False. For data sets with an even number of values, the median doesn’t have to be a value in the set — e.g. {1, 2, 2, 3, 3, 3} has a median 2.5
13. Not all data sets have a median.
True. A data set whose values cannot be put in order doesn’t have a median.
Solution follows…

21. Median and Mode 5.1.1 Round Up
Lesson
5.1.1
Median and Mode
Round Up
These typical values beginning with “m” can get confusing.
The median is the middle number when they’re arranged in order, and the mode is the most common value.
There’s another similar “m” coming up in the next Lesson too — the mean.