Name: _________________________________________________________ Period: ______ Date: ___________________________
11th Grade Mathematics PSSA Preparation Program
Mastered On: _____________________
Measures of Central Tendencies and Quartiles
Anchors Addressed
M11.E.2.1.1 – Calculate or select the appropriate measure of central tendency (mean, median, or mode) of a
set of data given or represented on a table, line plot, or stem-and-leaf plot.
M11.E.2.1.2 – Calculate and/or interpret the range, quartiles, and interquartile range of the data.
M11.E.2.1.3 – Describe how outliers affect measures of central tendency
Concepts
Measures of Central Tendency
Mean: average value of all of the data. To find the mean, add all of the values together and divide the sum by
the number of values in the data.
Median: the middle number in set of values arranged from least to greatest.
Mode: the number that appears most often in a set of data. (Data does not always have a mode.)
Example 1: Find the measures of central tendency for the following temperatures:
34, 26, 47, 12, 18, 22, 50, 37, 32, 23, 34
Solution: To find the mean add the values:
34 26 47 12 18 22 50 37 32 23 34 335
And then divide by the total number of values, 11.
335
30.45
11
To find the median, order the number from least to greatest:
12, 18, 22, 23, 26, 32, 34, 34, 37, 47, 50
Locate the number in the middle. If two numbers are in the middle, add then
together and divide by 2. The median is 32.
To find the mode, find the value(s) that occur the most. If two or more values have
the highest frequency, then record both the values. The mode of this data set is 34.
Quartiles and the Interquartile Range (IQR)
The word quartile is a fancy name for dividing the data into 4 equal parts. To find a quartile, find the median
and then find the median of each half of the data.
The interquartile range is a difference between the first and third quartiles. The diagram below represents
ordered data separated into quartiles.
D
A
T
Q2
median
Q1
A
Q3
Example 2: Find the interquartile range of the following data.
12, 17, 34, 56, 32, 23, 83, 31, 9, 36
Solution: To find the interquartile range, we first need to find each quartile by ordering the
numbers:
9, 12, 17, 23, 31, 32, 34, 36, 56, 83
Now, find the median and label it Q2. In this example, since the median is between
31 and 32, the median is:
31 32
31.5
2
Now, to find Q1 and Q3; to do this find the median of the two halves of the data:
9, 12, 17, 23, 31
32, 34, 36, 56, 83
Therefore, Q1 is 17 and Q3 is 36. Now, to find the IQR, subtract Q1 from Q3:
36 − 17 19
Therefore, the IQR is 19.
Exercises
A. Answer the following questions.
1. Explain how to use the median to find the quartiles.
B. Find the mean, median, and mode of the following data sets.
2. The sizes, in centimeters, of random perch caught in Lake Erie were:
12, 8, 7, 10, 6, 30, 23, 26, 18, 23, 14
Calculate the mean, median, and mode:
Mean:
Median:
Mode:
3. The points scored by a football team during each game from last season are:
14, 21, 18, 21, 14, 35, 42, 42, 21, 14
Calculate the mean, median, and mode:
Mean:
Median:
Mode:
C. Use the data provided to answer the questions below:
14, 9, 3, 11, 13, 16, 4, 11, 19, 10, 8, 20, 13, 7, 12
4. Write the data in order from least to greatest.
5. Find the median of the data set. This value is known as Quartile 2 or Q2 for short.
Q2:
6. Find the median of the first half of the data set. This value is known as Quartile 1 or Q1.
Q1:
7. Find the median of the second half of the data set. This value is known as Quartile 3 or Q3.
Q3:
8. Subtract the Q3 value from Q1. This is the Interquartile Range or IQR.
IQR:
Questions 4-8, above, represent the steps to finding the quartiles and the Interquartile Range. If the quartile
is between two data points, then just as when we find the median, add the two points together and divide by
two. Find Q1, Q2, Q3 and the IQR for the following data:
64, 82, 31, 43, 76, 55, 83, 59, 89
9. Write the data in order from least to greatest.
10. Find the median of the data set. This value is known as Quartile 2 or Q2 for short.
Q2:
11. Find the median of the first half of the data set. This value is known as Quartile 1 or Q1.
Q1:
12. Find the median of the second half of the data set. This value is known as Quartile 3 or Q3.
Q3:
13. Subtract the Q3 value from Q1. This is the Interquartile Range or IQR.
IQR:
D. Find the measures of central tendency, quartiles, and the interquartile range for the following data.
14. Data Set A: 64, 82, 31, 43, 76, 55, 83, 59, 89
15. Data Set B: 57, 65, 43, 72, 50, 62, 53, 49, 63, 55
E. Answer the following questions using the information provided.
pH is a scale that is used to determine the strength of acidity and alkalinity liquids. The scale ranges from 0
(being acidic) to 14 (being very alkalinity). In the middle of the scale is 7, which is neutral (neither acidic or
basic). The list below provides the pH of various liquids.
Liquid
Tap Water
Vinegar
Milk
Battery Acid
Ammonia
Lemon Juice
Bottled Water
Lye
Sea Water
Stomach Acid
Coca-Cola
pH
7.6
2.8
6.5
1.0
12.2
1.9
7.0
13.5
8.3
1.5
2.5
16. Determine the mean, median, and mode of the data.
17. Determine Q1, Q3, and the Interquartile Range.
18. Determine the minimum, maximum and range of the data. Recall the minimum is the smallest value
and the maximum is the largest value. The range is the difference between these values.
19. Of the calculations requested in questions 14-15, which calculation provides the most useful
information about the pH of these solutions? The least useful? Why?
20. Which calculation would best predict the pH of a solution that consisted of equal amounts of ALL of the
liquids in the list? Why?