2. 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?

3. Descriptive Statistics
Ungrouped Data
Measures
of Central
Tendency
Measures
of
Variability
Grouped Data
Measures
of Central
Tendency
Measures
of
Variability

4. PRE-ASSESSMENT
(Anticipation-Reaction Guide)
Before
Questions
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?
After

5. 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:
BENCH
PENSHOPPE
FOLDED & HUNG
SM DEPARTMENT STORE
You are to choose only one store. Which among the four
stores will you choose?

6. 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.

7. 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}

8. Pinklace sells ice cream. For five days, they sold
the following:
Number of cups of ice cream sold
Monday
13
Tuesday
27
Wednesday
15
Thursday
15
Friday
30
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?

9. 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.

10. 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.

11. 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:

12. Let’s Practice.
Find the mean of the following numbers:
1. 9, 15, 12, 10, 20
2. 100, 121, 132, 143
3. 54, 58, 61, 72, 81, 65

13. 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, 100
82
b. Mary: 90, 92, 95
63
Lina: 79, 80, 81
d. Willie: 65, 80, 80
c.
100
115

15. 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.

16. She made a list of the service crews and their ages:
Service Crew Age
Michelle
Sheryl
Karen
Mark
Jason
Oliver
Eliza
47
21
20
19
18
18
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.

17. Looking at the same list…
Service Crew
Michelle
Sheryl
Karen
Mark
Jason
Oliver
Eliza
Age
47
21
20
19
18
18
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?

18. 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.

19. 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, 10
The median is 8.

20. Find the median of the following sets of data:
32, 45, 22, 21, 18, 36, 50
2. 95, 95, 96, 88, 82, 100
3. 221, 332, 421, 326, 281, 220, 341, 109, 112
1.

21. 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. median
The average score on the last Pre-Algebra test was
85.
mean
The most common height on the basketball team is 6
ft 11 in. mode
One-half of the cars at a dealership cost less than
Php 700, 000.00. median
The average amount spent per customer in a
department store is Php 2, 500.00.
mean

22. 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 = 44
Mode = none
2. Mean = 47.6
Median = 49
Mode = 49
3. Mean = 38.33
Median = 34
Mode = 34

23. Solve the following problems.
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?
1.

24. 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?

25. 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.

26. 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.

27. The results (in months) are as follows:
Brand A:
Brand B:
10
35
60
45
50
30
30
35
40
40
20
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?

28. 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.

29. 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

30. Going back to the activity…
Brand A:
10 60
Largest Value = 60
50 30 40 20
Smallest Value = 10
RANGE = Largest Value – Smallest Value
= 60 – 10
= 50
Brand B:
35 45
Largest Value = 45
30 35 40 25
Smallest Value = 25
RANGE = Largest Value – Smallest Value
= 45 – 25
= 20

31. 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:
Factory B:
400, 450, 520, 380, 482, 495, 575, 450
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?

32. 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.

33. 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.

34. How to find the average/mean deviation:

35. Procedure in computing the average deviation
(Refer to the activity about wages of factory workers)
1. Find the mean of the scores.
Factory A:
400, 450, 520, 380, 482, 495, 575, 450
2. Find the absolute difference between each score and the mean.

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

37. 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

38. (Refer to the activity about wages of factory workers)
1. Find the mean of the scores.
Factory B:
450, 460, 462, 480, 450, 450, 400, 600
2. Find the absolute difference between each score and the mean.

39. 3. Find the sum of the absolute differences and then divide
by N.
Lower Average
Deviation means more
consistent scores.

40. 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

41. 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

42. The average deviation
gives a better approximate than
the range. However, it does not
lend
itself
readily
to
mathematical treatment for
deeper analysis.
It’s the
standard
deviation.
Then what measure
of variability is the
most reliable?

43. The Standard Deviation

1. Find the mean.

44. 2. Find the deviation from the mean.
39
5. Compute for the standard deviation.
289
10
22
-12
144
22
2
4
16
22
-6
36
19
4. Add all the squared deviations.
17
24
3. Square the deviations.
22
22
-3
9
26
22
4
16
29
22
7
49
30
22
8
64
5
22
-17
289
SUM 900

45. What does a
standard deviation
of 10 imply?
So does that mean
that a lower standard
deviation means less
varied scores?
It means that most of the
scores are within 10 units
from the mean.
That’s correct!
Lower standard deviation
shows more consistent
scores.

46. 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

47. Anticipation-Reaction Guide
Before
Questions
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?
After

49. 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
82
85
85
83
79
80
86
75
92
91
88
87
87
78
79
80
81
79
88
91
92
81
90
85
84
83
82
82
83
90
91
92
95
75
78
89
80
81
82
82
76
77
90
87
88
83
83
95
92
79
79
79
95
90
79
90
91
79
95
73
85
97
78
91
76

50. 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

51. RULES FOR GROUPING
1.
2.
3.
4.
5.
The intervals must cover the complete range of
values. The intervals need not begin nor end with
the lowest or highest values.
The intervals must be of equal size.
For effective grouping, the number of intervals
should be between 5 and 15.
Every score must be tallied from highest to lowest
or from lowest to highest.
Thus, the intervals should not overlap. When an
interval ends with a counting number, the next
intervals begins with the next counting number.

52. First
digits
(stem)
7
Second digits (leaf)
(0-4)
(5-9)
8
(0-4)
(5-9)
9
(0-4)
(5-9)

53. The Grouped Data
Grades
Frequency
70 – 74
75 – 79
80 – 84
85 – 89
90 – 94
95 – 99
1
17
18
15
16
5

54. Steps in Finding the Mean

55. Find the mean and the mode.
1
2
3
4
5
6
TOTAL
2
3
5
1
2
4

56. Find the mean , modal class, and the mode.
80 – 84
85 – 89
90 – 94
95 – 99
100 – 104
TOTAL
2
8
11
3
1

57. Steps in Finding the Median
1. Obtain the cumulative frequencies.

58. Classes
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

59. Mode
The mode is the midpoint of the
modal class.

60. RANGE
100 – 199
200 – 299
300 – 399
400 – 499
6
10
30
20

61. AVERAGE DEVIATION
Example: Calculate the mean deviation for the 30
marathon times in the grouped distribution as follows:
Time
(min)
Frequency
128-130
131-133
134-136
3
1
4
137-139 140-142
3
7
143-145
12

62. Finding the Average Deviation
Time (min)
Class Mark
(CM)
Frequency
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
30
4188 min
Sum

63. Time
(min)
Class
Mark
(CM)
Frequency fCM
f
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

64. Variance and Standard Deviation
x
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
95
5690
345680
SUM

65. THANK
YOU!