Measures Of Central Tendency And Measures Of Variability

Republic of the Philippines DEPARTMENT OF EDUCATION REGION IV (A) – CALABARZON Karangalan Village, Cainta, Rizal

REGIONAL MASS TRAINING OF GRADE 8 MATHEMATICS TEACHERS ON THE K TO 12 BASIC EDUCATION PROGRAM

MODULE 10: Measures of Central Tendency and Measures of Variability

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Lesson 1: Measures of Central Tendency of Ungrouped Data  Lesson 2: Measures of Variability of Ungrouped Data  Lesson 3: Measures of Central Tendency of Grouped Data o Lesson 4: Measures of Variability of Grouped Data 

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Objectives Lesson 1 At the end of the session, the students will be able to: 1. Find the mean, median, and mode of ungrouped data. 2. Describe and illustrate the mean, median and mode of ungrouped data.

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Objectives Lesson 2 3. Discuss the meaning of variability 4. Calculate the different measures of variability of a given ungrouped data; range; standard deviation and variance. 5. Describe and interpret data using measures of central tendency and measures of variablilty. [email protected] batangas

Think of This…

Think of a method of getting the sum of numbers (a) 1, 2, 3,…,10 (b) 1, 2, 3,…,17.

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Group Task Group 1 & 2 1. Given the following numbers 5, 4, 7, 8, 4, 6, 9 and 3 find the following: a. Mean b. Median c. Mode 2. Present your solution. [email protected] batangas

Group Task Groups 3 and 4 Karl got the following scores in seven 10-item Math quizzes: 5, 4, 9, 8, 5, 6 and 7. 1. Determine the following: a. Mean score b. Median score c. Modal score 2. Present your solution [email protected] batangas

Group Task Group 4 Given the following numbers: 14 11 1 22 12 15 20 6 7 13 23 27 9 12 14 17 16 3 18 5 8 4 24 20 30 1. Suggest a tabulation showing the groupings (by intervals) of these numbers. 2. Write each number on the interval where it belongs. 3. Present your work. [email protected] batangas

Group Task Group 5 Given the following numbers: 13 21 15 9 20

14

18,

a. Find the mean. b. Determine the difference of the mean and each given number. Tabulate your work. c. Present your work. [email protected] batangas

Let’s Discuss! For Group 1 & 2: 1. How did you find the activity? How do we call these values of Mean, Median and Mode? 2. How did you get the value of each measure of central tendency? Explain each. 3. Compare one value from another. Which two measures are of very close values? 4. Have you encountered any problem or difficulty in performing your task? Cite them. 5. How were you able to overcome such problem or difficulty? [email protected] batangas

Let’s Discuss! For Group 3 and 4: 1. How did you find the activity? 2. What is the value of each of Mean, Median and Mode? 3. How about the Mean Score, the Median Score and the Modal Score? Do you have the same answer with those of number 2 respectively? 4. Have you encountered any problem or difficulty in performing your task? Cite them. 5. How were you able to overcome such problem or difficulty? [email protected] batangas

Let’s Discuss! For Group 4:

1. How did you find the activity? 2. What is the basis of your interval used? How many groups were produced? 3. What do you think this activity imply? 4. Have you encountered any problem or difficulty in performing your task? Cite them. 5. How were you able to overcome such problem or difficulty? [email protected] batangas

Let’s Discuss! For Group 5: 1. How did you find the activity? 2. How far/close is the value of each number from the mean? Enumerate. 3. What do you think this value imply? 4. Have you encountered any problem or difficulty in performing your task? Cite them. 5. How were you able to overcome such problem or difficulty? [email protected] batangas

THE STANDARDS REVISITED... Grade 8 Mathematics Strand: Statistics MODULE 10: MEASURES OF CENTRAL TENDENCY AND MEASURES OF VARIABILITY

Learning Standard The learner demonstrates understanding and appreciation of key concepts and principles of mathematics as applied, using appropriate technology, in problem solving, critical thinking, communicating, reasoning, making connections, representations, and decisions in real life. [email protected] batangas

Grade Level Standard The learner demonstrates understanding of key concepts and principles of algebra, geometry, probability and statistics as applied, using appropriate technology, in critical thinking, problem solving, reasoning, communicating, making connections, representations, and decisions in real life.

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Content Standard

The learner demonstrates understanding of the key concepts of the different measures of tendency, variability of a given data, fundamental principles of counting and simple probability. Performance Standard The learner computes and applies accurately the descriptive measures in statistics to data analysis and interpretation in solving problems related to research, business, education, technology, science, economics, and others. [email protected] batangas

Learning Competencies The learner... 1. finds the mean, median and mode of statistical data. 2. describes and illustrates the mean, median and mode of ungrouped and grouped data. 3. discusses the meaning of variability. 4. calculates the different measures of variability of a given set of data: (a) range; (b) average deviation; (c) variance; (d) standard deviation. 5. describes a set of data using measures of central tendency and measures of variability. [email protected] batangas

DEFINITION/FORMULA OF MEASURES FOR UNGROUPED DATA

A. MEASURES OF CENTRAL TENDENCY

Mean (𝑥)

𝑥 𝑥= 𝑁 where Σx - the summation of x N - number of values of x.

Median (when data are arranged in ascending or descending values)

where N – number of given data [email protected] batangas

DEFINITION/FORMULA OF MEASURES FOR UNGROUPED DATA A. MEASURES OF CENTRAL TENDENCY

Mode It is the measure or value which occurs most frequently in a set of data. It is the value with the greatest frequency. To find the mode for a set of data: 1. select the measure that appears most often in the set; 2. if two or more measures appear the same number of times, then each of these values is a mode; and 3. if every measure appears the same number of times, then the set of data has no mode. [email protected] batangas

DEFINITION/FORMULA OF MEASURES FOR UNGROUPED DATA

B. MEASURES OF VARIABILITY

The Range (R) The range is the simplest measure of variability. R=H–L where R - Range H - Highest value L - Lowest value

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DEFINITION/FORMULA OF MEASURES FOR UNGROUPED DATA B. MEASURES OF VARIABILITY The Average Deviation The dispersion of a set of data about the average of these data is the average deviation or mean deviation. To compute the average deviation of an ungrouped data, we use the formula: 𝑥−𝑥 𝐴. 𝐷. = 𝑁 where A.D. is the average deviation x is the individual score 𝑥 is the mean N is the number of scores 𝑥 − 𝑥 is the absolute value of the deviation from the mean [email protected] batangas

DEFINITION/FORMULA OF MEASURES FOR UNGROUPED DATA

B. MEASURES OF VARIABILITY

The Standard Deviation 𝑥−𝑥 2 𝑆𝐷 = 𝑁−1 where SD is the standard deviation x is the individual score 𝑥 is the mean N is the number of scores [email protected] batangas

SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE

Consider a list of IQ scores for a gifted classroom in a particular elementary school. The IQ scores are: 118, 123, 124, 125, 127, 128, 129, 130, 130, 133, 136, 138, 141, 142, 149, 150, 154. [email protected] batangas

SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE Step 1: Figure out how many classes (categories) you need to consider the general guidelines: Use 5 if there are 17 to 32 data in the set, 6 if there are 33 to 64 data in the set, 7 if there are 65 to 128 data in the set, or in general, consider this: “The number of classes, k, should be the smallest integer such that 2k > n, where n is the number of observations.”

For the list of IQs above, we picked 5 classes because there are 17 given data. [email protected] batangas

SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE

Step 2: Subtract the minimum data value from the maximum data value. For example, our IQ list above had a minimum value of 118 and a maximum value of 154, so: 154 – 118 = 36

Step 3: Divide your answer in Step 2 by the number of classes you chose in Step 1. 36 / 5 = 7.2 Step 4: Round the number from Step 3 up to a whole number to get the class width or interval size. Rounded up, 7.2 becomes 8. [email protected] batangas

SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE

Step 5: Write down the lower limit of the first class interval. This should be the nearest number lower than the lowest value in the data set and a multiple of the class width. The lowest value is 118 and the nearest number lower than 118 which is a multiple of 8 is 112. Step 6: Add the class width from Step 4 to Step 5 to get the next lower class limit: 112 + 8 = 120 [email protected] batangas

SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE

Step 7: Repeat Step 6 for the other minimum data values (in other words, keep on adding your class width or interval size to your minimum data values) until you have created the number of classes you chose in Step 1. We chose 5 classes, so our 5 minimum data values are:

152 (144 + 8) 144 (136 + 8) 136 (128 + 8) 128 (120 + 8) 120 (112 + 8) 112 [email protected] batangas

SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE

Step 8: Write down the upper class limits. These are the highest values that can be in the category, so in most cases you can subtract 1 from class width and add that to the minimum data value. For example: (112 + 8) – 1 = 119 152 – 159 144 – 151 136 – 143 128 – 135 120 – 127 112 – 119 [email protected] batangas

SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE

Step 9: Add a second column for the number of items in each class, and label the columns with appropriate headings: IQ (CLASS INTERVAL) 152 – 159 144 – 151 136 – 143 128 – 135 120 – 127 112 – 119

Frequency

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SUGGESTED PROCEDURE IN PREPARING A FREQUENCY DISTRIBUTION TABLE Step 10: Count the number of items in each class, and put the total in the second column. The list of IQ scores are: 118, 123, 124, 125, 127, 128, 129, 130, 130, 133, 136, 138, 141, 142, 149, 150, 154.

IQ (CLASS INTERVAL) 152 – 159 144 – 151 136 – 143 128 – 135 120 – 127 112 – 119

Frequency | = 1 || = 2 |||| = 4 ||||| = 5 |||| = 4 | = 1 [email protected] batangas

DEFINITION/FORMULA OF MEASURES FOR GROUPED DATA

A. MEASURES OF CENTRAL TENDENCY

Mean To find the mean (𝑥 ) of grouped data using class marks, 𝑓𝑋 𝑥= 𝑓 where: f is the frequency of each class X is the class mark of the class

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an

DEFINITION/FORMULA OF MEASURES FOR GROUPED DATA To find the mean (𝑥 ) of grouped data using class marks, 𝑓𝑋 𝑥= 𝑓 A. MEASURES OF CENTRALf TENDENCY where: is the frequency of each class

Mean

X is the class mark of the class

Tomean find the mean (𝑥 ) of grouped datadeviation, using class mar To find the (𝑥 ) of grouped data using code 𝑓𝑑𝑓𝑋 𝑥 = 𝐴. 𝑀. +𝑥 = 𝑖 𝑓 𝑓 where: where:A.M. is the fassumed mean is the frequency of each cla f is the X frequency of each is the class markclass of the clas d is the coded deviation from A.M. i is the class interval

To find the mean (𝑥 ) of grouped data using code dev 𝑓𝑑 𝑥 = 𝐴. 𝑀. + 𝑖 [email protected] batangas

DEFINITION/FORMULA OF MEASURES FOR GROUPED DATA

Median OF CENTRAL TENDENCY A. MEASURES 𝑓 edian −<𝑐𝑓 2 Median 𝑓 𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙𝑏𝑚𝑐 + 𝑓𝑓 𝑖 −<𝑐𝑓 2 𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙𝑏𝑚𝑐 + 𝑖2 −<𝑚𝑐𝑐𝑓 𝑓𝑚𝑐+ 𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙𝑏𝑚𝑐 𝑖 𝑓𝑚𝑐 where: lbmc is the lower boundary of the median class; here: lbmc is the lower boundary of the median class; f is the frequency of each class; where: lbmcfrequency is the lower boundary of the median class; f is the of each class;
DEFINITION/FORMULA OF MEASURES FOR GROUPED DATA A. MEASURES OF CENTRAL TENDENCY Mode 𝑀𝑜𝑑𝑒 = 𝑙𝑏𝑚𝑜 + where:

𝐷1 𝐷1 +𝐷2

𝑖

lbmo is the lower boundary of the modal class; D1 is the difference between the frequencies of the modal class and the next upper class; D2 is the difference between the frequencies of the modal class and the next lower class; and i is the class interval.

The modal class is the class with the highest frequency. If there are two or more classes having the same highest frequency, the formula to be used is: [email protected] Mode = 3(Median) − 2(Mean)

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DEFINITION/FORMULA OF MEASURES FOR GROUPED DATA

B. MEASURES OF VARIABILITY

Range The range is the simplest measure of variability. Range = Upper Class Boundary – Lower Class Boundary of the Highest Interval of the Lowest Interval

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DEFINITION/FORMULA OF MEASURES FOR GROUPED DATA B. MEASURES OF VARIABILITY Variance (σ2) Variance is the average of the square deviation from the mean. For large quantities, the variance is computed using frequency distribution with columns for the midpoint value, the product of the frequency and midpoint value for each interval, the deviation and its square; and the product of the frequency and the squared deviation. To find variance of a grouped data, use the formula: 2 𝑓 𝑋 − 𝑥 𝜗2 = 𝑓−1 where: f is the class frequency; X is the class mark; 𝑥 is the class mean; and [email protected] 𝑓 is the total frequency of thebatangas median class.

DEFINITION/FORMULA OF MEASURES FOR GROUPED DATA B. MEASURES OF VARIABILITY

Standard Deviation (s) The standard deviation is considered the best indicator of the degree of dispersion among the measures of variability because it represents an average variability of the distribution. Given the set of data, the smaller the range, the smaller the standard deviation, the less spread is the distribution. 𝑠 = 𝜗2,

𝑠=

or

𝑓 𝑋−𝑥 𝑓−1

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What Can You Give? Share…

Open Discussion [email protected] batangas

?

It’s Your Turn… 1. Select a learning competency from Module 10. 2. Prepare five (5) questions of different levels: 2 Knowledge, 1 Process, 1 Understanding and 1 Product/ Performance. Make a rubric for the last item. 3. Submit your output. [email protected] batangas

Words to live by before you leave:

“Anuman ang pagkakaiba-iba at pagkakawatakwatak ng mga iniisip, saloobin at gawi ukol sa isang tunguhin, magtatapos din ito sa isang tuldok na may saysay, ang tuldok ng tagumpay, kung sa pangkalahatan ay Diyos ang sentro at gabay.” - Lee Yam

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