Measures of Central Tendencies and Quartiles

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1 Name: Period: Date: 11 th Grade Mathematics PSSA Preparation Program Mastered On: Measures of Central Tendencies and Quartiles Anchors Addressed M11.E 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 Calculate and/or interpret the range, quartiles, and interquartile range of the data. M11.E 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: And then divide by the total number of values, 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.

2 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 A Example 2: Find the interquartile range of the following data. Q1 Q2 median 12, 17, 34, 56, 32, 23, 83, 31, 9, 36 Q3 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: 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: Therefore, the IQR is Exercises A. Answer the following questions. 1. Explain how to use the median to find the quartiles.

3 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, 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:

4 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: 9. Write the data in order from least to greatest. 64, 82, 31, 43, 76, 55, 83, 59, 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, Data Set B: 57, 65, 43, 72, 50, 62, 53, 49, 63, 55

5 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 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?