measures of central tendency and variability

Measures of Central Tendency and Variability

Prepared by MARIELE D. BRUTAS

Introduction and Focus Questions

Have you ever wondered why a certain size of shoe or brand of shirt is made more available than other sizes?

Have you ever asked yourself why a certain basketball player gets more playing time than the rest of his team mates?

Have you ever thought of comparing your academic performance with your classmates? Have you ever wondered what score you need for each subject to qualify for honors? Have you at certain time asked yourself how norms and standards are made?

Descriptive Statistics

Ungrouped Data

Measures of

Central Tendenc

y

Measures of

Variability

Grouped Data

Measures of

Central Tendenc

y

Measures of

Variability

PRE-ASSESSMENT (Anticipation-Reaction Guide)

Before Questions After

Which measure of central tendency is generally used in determining the size of the most saleable shoes in a department store?

What is the most reliable measure of variability?

Which measure of central tendency is greatly affected by extreme scores?

Margie has grades 86, 68, and 79 in her first three tests in Algebra. What grade must she obtain on the 4th test to get an average of 78?

What is the median age of a group of employees whose ages are 36, 38, 24, 21, and 27?

If the range of a set of scores is 14 and the lowest score is 7, what is the highest score?

What is the standard deviation of the scores 5, 4, 3, 6, and 2?

The Situation…

You are one of the winners in a contest where the prizes are gift certificates from the famous stores in the city. The sponsors are the following:

BENCHPENSHOPPEFOLDED & HUNGSM DEPARTMENT STORE

You are to choose only one store. Which among the four stores will you choose?

The Mode

The mode is the value or element which occurs most frequently in a set of data. It is the value or element with the greatest frequency. Mode can be quantitative or qualitative. 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;3. if every measure appears the same number of times, then the set of data has no mode.

Try answering these items…

Find the mode in the given sets of scores.1. {10, 12, 9, 10, 13, 11, 10}2. {15, 20, 18, 19, 18, 16, 20, 18}3. {5, 8, 7, 9, 6, 8, 5}4. {7, 10, 8, 5, 9, 6, 4}5. {12, 16, 14, 15, 16, 13, 14}6. {smart, globe, sun, sun, sun, globe, smart}7. {1, 1, 1, 2, 2, 2, 3, 3, 3}

Pinklace sells ice cream. For five days, they sold the following:

Questions:1. What is the total number of cups of ice cream sold

during the whole week?2. If Pinklace will be able to sell same number of cups of

ice cream each day, how many will it be?

Number of cups of ice cream sold

Monday 13

Tuesday 27

Wednesday 15

Thursday 15

Friday 30

The Mean

The mean (also known as arithmetic mean) is the most commonly used measure of central position. It is used to describe a set of data where the measures cluster or concentrate at a point. As the measures cluster around each other, a single value appears to represent distinctively the typical value.

How do we compute for the mean?

It is the sum of measures x divided by the number N of measures. It is symbolized as x (read as x bar). To find the mean of an ungrouped data, use the formula

where = summation of x (sum of measures) and

N= number of values of x.

𝑚𝑒𝑎𝑛 ( 𝑥 )=∑ 𝑥𝑁

Let’s practice…

The grades in Geometry of 10 students are 87, 84, 85, 85, 86, 90, 79, 82, 78, 76. What is the average grade of 10 students?

Solution:

Mean(

Let’s Practice.

Find the mean of the following numbers:

1. 9, 15, 12, 10, 202. 100, 121, 132, 1433. 54, 58, 61, 72, 81, 65

WORK IN PAIRS

The first three test scores of each of the four students are shown. Each student hopes to maintain an average of 85. Find the score needed by each student on the fourth test to have an average of 85, or explain why such average is not possible.

a. Lisa: 78, 80, 100b. Mary: 90, 92, 95c. Lina: 79, 80, 81d. Willie: 65, 80, 80

82

63

100

115

Who’s Representing

?

The situation…

Sonya’s Kitchen received an invitation in a Food Exposition. All the seven service crew are eager to go but only one can represent the restaurant. To be fair, Sonya thought of sending the crew whose age is in the middle of the ages of the seven crews.

She made a list of the service crews and their ages:

Service Crew Age

Michelle 47Sheryl 21Karen 20Mark 19Jason 18Oliver 18Eliza 18

Guide Questions:

1. What is the mean age of the service crew?

2. Is there someone in this group who has this age?

3. How many persons are older than the mean age? How many are younger?

4. Do you think this is the best measure of central tendency to use? Explain.

Looking at the same list…

Service Crew Age

Michelle 47Sheryl 21Karen 20Mark 19Jason 18Oliver 18Eliza 18

Guide Questions:

1. Arrange the ages in numerical order.

2. What is the middle value?3. Is there a crew with this

representative age?4. How many crew are younger

than this age? Older than this age?

5. Compare the result with the previous activity. Which result do you think is a better basis of choosing the representative?

6. Who is now the representative of Sonya’s Kitchen in the Food Fair?

The Median

The median is the middlemost value or term in a set of data arranged according to size/magnitude (either increasing or decreasing). If the number of values is even, the median is the average of the two middlemost values.

Let’s practice…

Andrea’s scores in 9 quizzes during the first quarter are 8, 7, 6, 10, 9, 5, 9, 6, and 10. Find

the median.

Solution

Arrange the scores in increasing order.5, 6, 6, 7, 8, 9, 9, 10, 10The median is 8.

Find the median of the following sets of data:

1. 32, 45, 22, 21, 18, 36, 502. 95, 95, 96, 88, 82, 1003. 221, 332, 421, 326, 281, 220, 341, 109, 112

What measure of central tendency is used in the following situations?

Kevin noticed that half of the cereal brands in the store cost more than Php 150.00.

The average score on the last Pre-Algebra test was 85.

The most common height on the basketball team is 6 ft 11 in.

One-half of the cars at a dealership cost less than Php 700, 000.00.

The average amount spent per customer in a department store is Php 2, 500.00.

median

mean

mode

median

mean

Calculate the mean, median, and mode of each set of numbers.

1. 4, 14, 29, 44, 46, 52, 55

2. 42, 49, 49, 49, 49

3. 22, 34, 34, 34, 45, 61

4. 20, 22, 56, 62, 63, 67

5. 11, 33, 54, 54, 71, 84, 93

1. Mean = 34. 86

Median = 44Mode = none

2. Mean = 47.6Median = 49Mode = 49

3. Mean = 38.33Median = 34Mode = 34

Solve the following problems.

1. Andy has grades of 84, 65, and 76 on three math tests.  What grade must he obtain on the next test to have an average of exactly 80 for the four tests?

2. A storeowner kept a tally of the sizes of suits purchased in her store. Which measure of central tendency should the storeowner use to describe the most saleable suit?

3. A tally was made of the number of times each color of crayon was used by a kindergarten class.  Which measure of central tendency should the teacher use to determine which color is the favorite color of her class?

Continuation…

4. In January of 2006, your family moved to a tropical climate.  For the year that followed, you recorded the number of rainy days that occurred each month.  Your data contained 14, 14, 10, 12, 11, 13, 11, 11, 14, 10, 13, 12.a.  Find the mean, mode, and the median for your

data set of rainy days.b.  If the number of rainy days doubles each month in

the year 2007, what will be the mean, mode, median?c.  If, instead, there are three more rainy days per

month in the year 2007, what will be the mean, mode, median?

Continuation…

5. The values of 11 houses on Washington Street are shown in the table.

a.  Find the mean value of these houses in dollars.b.  Find the median value of these houses in dollars.c.  State which measure of central tendency, the mean

or the median, best represents the values of these 11 houses.  Justify your

answer.

The situation…

A testing laboratory wishes to test two experimental brands of outdoor paint to see how long each paint will last before fading. The testing lab makes use of six gallons of paint for each brand name to test.

The results (in months) are as follows:

Brand A: 10 60 50 30 40 20Brand B: 35 45 30 35 40 25

Guide Questions:1. What is the mean score of each brand?2. Can the mean of each brand be a good basis for

comparing them?3. Which brand has results closer to the mean? 4. If you are to choose from these two brands,

which would you prefer? Why?

Measures of Dispersion or Variability

-refer to the spread of the values about the mean. These are important quantities used by statisticians in evaluation. Smaller dispersion of scores arising from the comparison often indicates more consistency and more reliability.

The most commonly used measures of dispersion are the range, the average deviation, the standard deviation, and variance.

The Range

The range is the simplest measure of variability. It is the difference between the largest value and the smallest value.

Range= Largest Value – Smallest Value

Going back to the activity…

Brand A: 10 60 50 30 40 20

Brand B: 35 45 30 35 40 25

Largest Value = 60

Smallest Value = 10

Largest Value = 45

Smallest Value = 25

RANGE = Largest Value – Smallest Value= 60 – 10

= 50

RANGE = Largest Value – Smallest Value = 45 – 25

= 20

MO

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Now consider another situation

The following are the daily wages of 8 factory workers of two garments factories A and B. Find the range of salaries in peso (Php).

Factory A: 400, 450, 520, 380, 482, 495, 575, 450Factory B: 450, 460, 462, 480, 450, 450, 400, 600

Questions:1. What is the mean wage of each group of workers?2. What is the range of wages of each group of workers?3. For this case, are you convinced that the group with lower range

has more consistent wages?

Though the range is the simplest and easiest to find measure of variability, it is not a stable measure. Its value can fluctuate greatly even with a change in just a single value, either the highest or lowest.

The Average/Mean Deviation

The dispersion of a set of data about the average of these data is the average deviation or the mean deviation.

How to find the average/mean deviation:

𝑨 .𝑫 .=∑|𝒙 −𝒙|

𝑵

where A.D. is the average deviation,is the individual scoreis the meanis the number of scores

is the absolute value of the deviation from the mean

Procedure in computing the average deviation(Refer to the activity about wages of factory workers)

Factory A: 400, 450, 520, 380, 482, 495, 575, 450

¿|450−469|=19

¿|380−469|=89

|𝒙 −𝒙| |𝒙 −𝒙|

1. Find the mean of the scores.

mean () =

2. Find the absolute difference between each score and the mean.

3. Find the sum of the absolute differences and then divide by N.

= 49

or you may present using a table…

400 469 -69 69

450 469 -19 19

520 469 51 51

380 469 -89 89

482 469 13 13

495 469 26 26

575 469 106 106

450 469 -19 19

Total 392

(Refer to the activity about wages of factory workers)

Factory B: 450, 460, 462, 480, 450, 450, 400, 600

¿|4 60−469|=9

¿|480−469|=11

|𝒙 −𝒙| |𝒙 −𝒙|

1. Find the mean of the scores.

mean () =

2. Find the absolute difference between each score and the mean.

3. Find the sum of the absolute differences and then divide by N.

Lower Average Deviation means more consistent

scores.

450 469 -19 19

460 469 -9 9

462 469 -7 7

480 469 11 11

450 469 -19 19

450 469 -19 19

400 469 -69 69

600 469 131 131

Total 284

Find the Average Deviation of the following:

1. Science Achievement Scores:

60, 75, 80, 85, 90, 95

2. The weights in kilogram of 10 students are: 52, 55, 50, 55, 43, 45, 40, 48, 45, and 47

The average deviation gives a better approximate than the range. However, it does not lend itself readily to mathematical treatment for deeper analysis.

Then what measure of

variability is the most reliable?

It’s the standard deviation.

The Standard Deviation

Compute the standard deviation of the set of test scores:

1. Find the mean.

2. Find the deviation from the mean.

3. Square the deviations.

4. Add all the squared deviations.

39 22 17 289

10 22 -12 144

24 22 2 4

16 22 -6 36

19 22 -3 9

26 22 4 16

29 22 7 49

30 22 8 64

5 22 -17 289

SUM 900

5. Compute for the standard deviation.

𝑆𝐷=√∑ ( 𝑥−𝑥 )2

𝑁=√ 9009 =𝟏𝟎

What does a standard

deviation of 10 imply?

It means that most of the scores are within 10 units from the mean.

So does that mean that a lower

standard deviation means less varied

scores?

That’s correct!Lower standard

deviation shows more consistent scores.

Let’s practice…

Compare the standard deviation of the scores of the three students in their Mathematics quizzes.

Student Mathematics Quizzes

A 97, 92, 96, 95, 90

B 94, 94, 92, 94, 96

C 95. 94, 93, 96, 92

Anticipation-Reaction GuideBefore Questions After

Which measure of central tendency is generally used in determining the size of the most saleable shoes in a department store?

What is the most reliable measure of variability?

Which measure of central tendency is greatly affected by extreme scores?

Margie has grades 86, 68, and 79 in her first three tests in Algebra. What grade must she obtain on the 4th test to get an average of 78?

What is the median age of a group of employees whose ages are 36, 38, 24, 21, and 27?

If the range of a set of scores is 14 and the lowest score is 7, what is the highest score?

What is the standard deviation of the scores 5, 4, 3, 6, and 2?

GROUPED DATA

You were asked to find the mean, median and mode of the Math grades of all the students in 2 Learning Groups.

The grades of 72 students are as follows:85 78 82 88 89 92 90 79 7982 85 85 83 79 80 86 79 9575 92 91 88 87 87 78 79 7379 80 81 79 88 91 92 95 8581 90 85 84 83 82 82 90 9783 90 91 92 95 75 78 79 7889 80 81 82 82 76 77 90 9190 87 88 83 83 95 92 91 76

Organizing data

STEM-LEAF DIAGRAM

First digits (stem)

Second digits (leaf)

7

8

9

Grades Frequency

70-79 18

80-89 33

90-99 21

GROUPED DATA

RULES FOR GROUPING

1. The intervals must cover the complete range of values. The intervals need not begin nor end with the lowest or highest values.

2. The intervals must be of equal size.3. For effective grouping, the number of

intervals should be between 5 and 15.4. Every score must be tallied from highest

to lowest or from lowest to highest.5. Thus, the intervals should not overlap.

When an interval ends with a counting number, the next intervals begins with the next counting number.

First digits (stem)

Second digits (leaf)

7 (0-4)

(5-9)

8 (0-4)

(5-9)

9 (0-4)

(5-9)

The Grouped Data

Grades Frequency

70 – 74 175 – 79 1780 – 84 1885 – 89 1590 – 94 1695 – 99 5

Steps in Finding the Mean

1. Compute for the Class Mark (CM).

2. Add the frequencies to obtain .

3. Compute for the values of CM by multiplying each class mark by the corresponding frequency.

4. Add the values of CM to obtain

5. Substitute in the values in the formula for the mean:

Find the mean and the mode.

Score Frequency )

1 2  2 3  3 5  4 1  5 2  6 4  

TOTAL    

Find the mean , modal class, and the mode.

Class Frequency

Class Mark

80 – 84 2    

85 – 89 8    

90 – 94 11    

95 – 99 3    

100 – 104 1    

TOTAL      

Steps in Finding the Median

1. Obtain the cumulative frequencies.

2. Identify the class that contains the median and obtain the median location .

3. Apply the formula:

Where = lower boundary of the class containing the median = the cumulative frequency of the class preceding the class

containing the median = the number of observations in the class containing the

median = size of class interval

Classes Cumulative Frequency ()

Position of the Data

1 – 4 16 16 1 – 16

5 – 8 20 36 17 – 36

9 – 12 28 64 37 – 64

13 – 16 24 88 65 – 88

17 – 20 16 104 89 – 104

21 – 24 11 115 105 – 115

25 – 28 5 120 116 – 120

TOTAL 120

Mode

The mode is the midpoint of the modal class.

RANGE

𝑹𝒂𝒏𝒈𝒆=𝒖𝒑𝒑𝒆𝒓 𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒚𝒐𝒇 𝒉𝒊𝒈𝒉𝒆𝒔𝒕 𝒄𝒍𝒂𝒔𝒔− 𝒍𝒐𝒘𝒆𝒓 𝒃𝒐𝒖𝒏𝒅𝒂𝒓𝒚 𝒐𝒇 𝒍𝒐𝒘𝒆𝒓 𝒄𝒍𝒂𝒔𝒔

100 – 199 200 – 299 300 – 399 400 – 499

6 10 30 20

𝑅𝑎𝑛𝑔𝑒=499.5−99.5=𝟒𝟎𝟎

AVERAGE DEVIATION

Example: Calculate the mean deviation for the 30 marathon times in the grouped distribution as follows:

Time (min)

128-130

131-133

134-136

137-139

140-142

143-145

Frequency 3 1 4 3 7 12

Finding the Average Deviation

Time (min) Class Mark (CM)

Frequency (min)

128 – 130 129 3 387

131 – 133 130 1 132

134 – 136 135 4 540

137 – 139 138 3 414

140 – 142 141 7 987

143 – 145 144 12 1728

Sum 30 4188 min

min

Time (min)

Class Mark (CM)

Frequency

f

f CM

128-130 129 3 387 10.6 31.8

131-133 132 1 132 7.6 7.6

134-136 135 4 540 4.6 18.4

137-139 138 3 414 1.6 4.8

140-142 141 7 987 1.4 9.8

143-145 144 12 1728 4.4 52.8

30 4188 125.2 min

𝐴 .𝐷 . ≈∑ 𝑓 |𝐶𝑀 −𝜇|

𝑁=125.2

30=4.17333min .∨4.2min .

Variance and Standard Deviationx Mid-class value, Frequency,

40-44 42 1 42 1764

45-49 47 7 329 15463

50-54 52 12 624 32448

55-59 57 24 1368 77976

60-64 62 29 1798 111476

65-69 67 14 938 62846

70-74 72 5 360 25920

75-79 77 3 231 17787

SUM 95 5690 345680

THANK YOU!