Measures of Central Tendency (Mean) | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year PDF Download

 Page 1


MeasuresofCentralTendency: FormulasandExamples
This document provides the essential formulas for computing the mean, median, and mode for
ungroupedandgroupeddata,withmultipleexamplestoillustrateeachmethod. Abrieftheoret-
ical overview is included at the end to explain their purpose and applicability. All calculations
are veri?edfor accuracy.
Mean
Themean(arithmeticaverage)isthesumofallscoresdividedbythenumberofscores,denoted
byx fora sample andµ for a population.
UngroupedData
Formula:
x =
?
x
N
where
?
x isthe sum of all scores, andN isthe number of scores.
Example1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39 (N = 10).
?
x = 58+34+32+47+74+67+35+34+30+39 = 450
x =
450
10
= 45
The mean is 45.
Example2: Heights(cm): 165, 170, 168, 172, 175 (N = 5).
?
x = 165+170+168+172+175 = 850
x =
850
5
= 170
Themean is 170 cm.
GroupedData
Formula:
x =
?
fx
N
where f is the frequency, x is the midpoint of each class interval, and N =
?
f is the total
number of scores.
Example1: Datawith class intervali = 5:
1
Page 2


MeasuresofCentralTendency: FormulasandExamples
This document provides the essential formulas for computing the mean, median, and mode for
ungroupedandgroupeddata,withmultipleexamplestoillustrateeachmethod. Abrieftheoret-
ical overview is included at the end to explain their purpose and applicability. All calculations
are veri?edfor accuracy.
Mean
Themean(arithmeticaverage)isthesumofallscoresdividedbythenumberofscores,denoted
byx fora sample andµ for a population.
UngroupedData
Formula:
x =
?
x
N
where
?
x isthe sum of all scores, andN isthe number of scores.
Example1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39 (N = 10).
?
x = 58+34+32+47+74+67+35+34+30+39 = 450
x =
450
10
= 45
The mean is 45.
Example2: Heights(cm): 165, 170, 168, 172, 175 (N = 5).
?
x = 165+170+168+172+175 = 850
x =
850
5
= 170
Themean is 170 cm.
GroupedData
Formula:
x =
?
fx
N
where f is the frequency, x is the midpoint of each class interval, and N =
?
f is the total
number of scores.
Example1: Datawith class intervali = 5:
1
Class Interval Frequency(f) Midpoint (x) fx
35–39 5 37 185
30–34 7 32 224
25–29 5 27 135
20–24 6 22 132
15–19 4 17 68
10–14 3 12 36
Total N = 30
?
fx = 780
x =
780
30
= 26
The mean is 26.
Example2: Testscores with class intervali = 10:
Class Interval Frequency(f) Midpoint (x) fx
80–89 4 84.5 338
70–79 6 74.5 447
60–69 8 64.5 516
50–59 5 54.5 272.5
40–49 2 44.5 89
Total N = 25
?
fx = 1662.5
x =
1662.5
25
= 66.5
The mean is 66.5.
GroupedData(ShortcutMethodwithAssumedMean)
Formula:
x =AM +
(?
fx
'
N
×i
)
where AM is the assumed mean (chosen near the distribution’s center), x
'
=
x-AM
i
, i is the
class interval,andN =
?
f.
Example1: Usingthe ?rst grouped data example,assumeAM = 22 (i = 5):
Class Interval f x x
'
=
x-22
5
fx
'
35–39 5 37 3 15
30–34 7 32 2 14
25–29 5 27 1 5
20–24 6 22 0 0
15–19 4 17 -1 -4
10–14 3 12 -2 -6
Total N = 30
?
fx
'
= 24
2
Page 3


MeasuresofCentralTendency: FormulasandExamples
This document provides the essential formulas for computing the mean, median, and mode for
ungroupedandgroupeddata,withmultipleexamplestoillustrateeachmethod. Abrieftheoret-
ical overview is included at the end to explain their purpose and applicability. All calculations
are veri?edfor accuracy.
Mean
Themean(arithmeticaverage)isthesumofallscoresdividedbythenumberofscores,denoted
byx fora sample andµ for a population.
UngroupedData
Formula:
x =
?
x
N
where
?
x isthe sum of all scores, andN isthe number of scores.
Example1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39 (N = 10).
?
x = 58+34+32+47+74+67+35+34+30+39 = 450
x =
450
10
= 45
The mean is 45.
Example2: Heights(cm): 165, 170, 168, 172, 175 (N = 5).
?
x = 165+170+168+172+175 = 850
x =
850
5
= 170
Themean is 170 cm.
GroupedData
Formula:
x =
?
fx
N
where f is the frequency, x is the midpoint of each class interval, and N =
?
f is the total
number of scores.
Example1: Datawith class intervali = 5:
1
Class Interval Frequency(f) Midpoint (x) fx
35–39 5 37 185
30–34 7 32 224
25–29 5 27 135
20–24 6 22 132
15–19 4 17 68
10–14 3 12 36
Total N = 30
?
fx = 780
x =
780
30
= 26
The mean is 26.
Example2: Testscores with class intervali = 10:
Class Interval Frequency(f) Midpoint (x) fx
80–89 4 84.5 338
70–79 6 74.5 447
60–69 8 64.5 516
50–59 5 54.5 272.5
40–49 2 44.5 89
Total N = 25
?
fx = 1662.5
x =
1662.5
25
= 66.5
The mean is 66.5.
GroupedData(ShortcutMethodwithAssumedMean)
Formula:
x =AM +
(?
fx
'
N
×i
)
where AM is the assumed mean (chosen near the distribution’s center), x
'
=
x-AM
i
, i is the
class interval,andN =
?
f.
Example1: Usingthe ?rst grouped data example,assumeAM = 22 (i = 5):
Class Interval f x x
'
=
x-22
5
fx
'
35–39 5 37 3 15
30–34 7 32 2 14
25–29 5 27 1 5
20–24 6 22 0 0
15–19 4 17 -1 -4
10–14 3 12 -2 -6
Total N = 30
?
fx
'
= 24
2
x = 22+
(
24
30
×5
)
= 22+4 = 26
Themean is 26, matching the direct method.
Example2: Usingthe second grouped data example,assumeAM = 64.5 (i = 10):
Class Interval f x x
'
=
x-64.5
10
fx
'
80–89 4 84.5 2 8
70–79 6 74.5 1 6
60–69 8 64.5 0 0
50–59 5 54.5 -1 -5
40–49 2 44.5 -2 -4
Total N = 25
?
fx
'
= 5
x = 64.5+
(
5
25
×10
)
= 64.5+2 = 66.5
The mean is 66.5, matching the direct method.
Median
The median is the middle value in an ordered dataset, denoted by M
d
. For grouped data, it
assumes linear interpolation within the median class.
UngroupedData(OddNumberofScores)
Formula:
M
d
= valueof the
(
N +1
2
)
th
score
Example 1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30 (N = 9). Ordered: 30, 32, 34, 34, 35,
47, 58, 67, 74.
N +1
2
=
9+1
2
= 5
The 5th score is 35, soM
d
= 35.
Example2: Weights(kg): 60, 65, 70, 55, 68 (N = 5). Ordered: 55, 60, 65, 68, 70.
N +1
2
=
5+1
2
= 3
The 3rd score is 65, soM
d
= 65.
UngroupedData(EvenNumberofScores)
Formula:
M
d
=
valueof
(
N
2
)
th
score +valueof
(
N
2
+1
)
th
score
2
3
Page 4


MeasuresofCentralTendency: FormulasandExamples
This document provides the essential formulas for computing the mean, median, and mode for
ungroupedandgroupeddata,withmultipleexamplestoillustrateeachmethod. Abrieftheoret-
ical overview is included at the end to explain their purpose and applicability. All calculations
are veri?edfor accuracy.
Mean
Themean(arithmeticaverage)isthesumofallscoresdividedbythenumberofscores,denoted
byx fora sample andµ for a population.
UngroupedData
Formula:
x =
?
x
N
where
?
x isthe sum of all scores, andN isthe number of scores.
Example1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39 (N = 10).
?
x = 58+34+32+47+74+67+35+34+30+39 = 450
x =
450
10
= 45
The mean is 45.
Example2: Heights(cm): 165, 170, 168, 172, 175 (N = 5).
?
x = 165+170+168+172+175 = 850
x =
850
5
= 170
Themean is 170 cm.
GroupedData
Formula:
x =
?
fx
N
where f is the frequency, x is the midpoint of each class interval, and N =
?
f is the total
number of scores.
Example1: Datawith class intervali = 5:
1
Class Interval Frequency(f) Midpoint (x) fx
35–39 5 37 185
30–34 7 32 224
25–29 5 27 135
20–24 6 22 132
15–19 4 17 68
10–14 3 12 36
Total N = 30
?
fx = 780
x =
780
30
= 26
The mean is 26.
Example2: Testscores with class intervali = 10:
Class Interval Frequency(f) Midpoint (x) fx
80–89 4 84.5 338
70–79 6 74.5 447
60–69 8 64.5 516
50–59 5 54.5 272.5
40–49 2 44.5 89
Total N = 25
?
fx = 1662.5
x =
1662.5
25
= 66.5
The mean is 66.5.
GroupedData(ShortcutMethodwithAssumedMean)
Formula:
x =AM +
(?
fx
'
N
×i
)
where AM is the assumed mean (chosen near the distribution’s center), x
'
=
x-AM
i
, i is the
class interval,andN =
?
f.
Example1: Usingthe ?rst grouped data example,assumeAM = 22 (i = 5):
Class Interval f x x
'
=
x-22
5
fx
'
35–39 5 37 3 15
30–34 7 32 2 14
25–29 5 27 1 5
20–24 6 22 0 0
15–19 4 17 -1 -4
10–14 3 12 -2 -6
Total N = 30
?
fx
'
= 24
2
x = 22+
(
24
30
×5
)
= 22+4 = 26
Themean is 26, matching the direct method.
Example2: Usingthe second grouped data example,assumeAM = 64.5 (i = 10):
Class Interval f x x
'
=
x-64.5
10
fx
'
80–89 4 84.5 2 8
70–79 6 74.5 1 6
60–69 8 64.5 0 0
50–59 5 54.5 -1 -5
40–49 2 44.5 -2 -4
Total N = 25
?
fx
'
= 5
x = 64.5+
(
5
25
×10
)
= 64.5+2 = 66.5
The mean is 66.5, matching the direct method.
Median
The median is the middle value in an ordered dataset, denoted by M
d
. For grouped data, it
assumes linear interpolation within the median class.
UngroupedData(OddNumberofScores)
Formula:
M
d
= valueof the
(
N +1
2
)
th
score
Example 1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30 (N = 9). Ordered: 30, 32, 34, 34, 35,
47, 58, 67, 74.
N +1
2
=
9+1
2
= 5
The 5th score is 35, soM
d
= 35.
Example2: Weights(kg): 60, 65, 70, 55, 68 (N = 5). Ordered: 55, 60, 65, 68, 70.
N +1
2
=
5+1
2
= 3
The 3rd score is 65, soM
d
= 65.
UngroupedData(EvenNumberofScores)
Formula:
M
d
=
valueof
(
N
2
)
th
score +valueof
(
N
2
+1
)
th
score
2
3
Example1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39 (N = 10). Ordered: 30, 32, 34, 34,
35, 39, 47, 58, 67, 74.
M
d
=
35+39
2
= 37
The median is 37.
Example2: Ages: 25, 30, 28, 35, 27, 32 (N = 6). Ordered: 25, 27, 28, 30, 32, 35.
M
d
=
28+30
2
= 29
The median is 29.
GroupedData
Formula:
M
d
=L+
[
N
2
-F
f
m
]
×i
whereListhelowerlimitofthemedianclass,F isthecumulativefrequencybeforethemedian
class,f
m
is the frequencyof the median class, andi isthe class interval.
Example1: DatawithN = 30:
ClassInterval Frequency(f)
35–39 5
30–34 7
25–29 5
20–24 6
15–19 4
10–14 3
Total N = 30
-
N
2
= 15. Cumulative frequencies: 10–14: 3; 15–19: 7; 20–24: 13; 25–29: 18. Median class:
25–29 (L = 24.5,f
m
= 5,F = 13,i = 5).
M
d
= 24.5+
[
15-13
5
]
×5 = 24.5+2 = 26.5
Themedian is 26.5.
Example2: DatawithN = 25:
ClassInterval Frequency(f)
80–89 4
70–79 6
60–69 8
50–59 5
40–49 2
Total N = 25
4
Page 5


MeasuresofCentralTendency: FormulasandExamples
This document provides the essential formulas for computing the mean, median, and mode for
ungroupedandgroupeddata,withmultipleexamplestoillustrateeachmethod. Abrieftheoret-
ical overview is included at the end to explain their purpose and applicability. All calculations
are veri?edfor accuracy.
Mean
Themean(arithmeticaverage)isthesumofallscoresdividedbythenumberofscores,denoted
byx fora sample andµ for a population.
UngroupedData
Formula:
x =
?
x
N
where
?
x isthe sum of all scores, andN isthe number of scores.
Example1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39 (N = 10).
?
x = 58+34+32+47+74+67+35+34+30+39 = 450
x =
450
10
= 45
The mean is 45.
Example2: Heights(cm): 165, 170, 168, 172, 175 (N = 5).
?
x = 165+170+168+172+175 = 850
x =
850
5
= 170
Themean is 170 cm.
GroupedData
Formula:
x =
?
fx
N
where f is the frequency, x is the midpoint of each class interval, and N =
?
f is the total
number of scores.
Example1: Datawith class intervali = 5:
1
Class Interval Frequency(f) Midpoint (x) fx
35–39 5 37 185
30–34 7 32 224
25–29 5 27 135
20–24 6 22 132
15–19 4 17 68
10–14 3 12 36
Total N = 30
?
fx = 780
x =
780
30
= 26
The mean is 26.
Example2: Testscores with class intervali = 10:
Class Interval Frequency(f) Midpoint (x) fx
80–89 4 84.5 338
70–79 6 74.5 447
60–69 8 64.5 516
50–59 5 54.5 272.5
40–49 2 44.5 89
Total N = 25
?
fx = 1662.5
x =
1662.5
25
= 66.5
The mean is 66.5.
GroupedData(ShortcutMethodwithAssumedMean)
Formula:
x =AM +
(?
fx
'
N
×i
)
where AM is the assumed mean (chosen near the distribution’s center), x
'
=
x-AM
i
, i is the
class interval,andN =
?
f.
Example1: Usingthe ?rst grouped data example,assumeAM = 22 (i = 5):
Class Interval f x x
'
=
x-22
5
fx
'
35–39 5 37 3 15
30–34 7 32 2 14
25–29 5 27 1 5
20–24 6 22 0 0
15–19 4 17 -1 -4
10–14 3 12 -2 -6
Total N = 30
?
fx
'
= 24
2
x = 22+
(
24
30
×5
)
= 22+4 = 26
Themean is 26, matching the direct method.
Example2: Usingthe second grouped data example,assumeAM = 64.5 (i = 10):
Class Interval f x x
'
=
x-64.5
10
fx
'
80–89 4 84.5 2 8
70–79 6 74.5 1 6
60–69 8 64.5 0 0
50–59 5 54.5 -1 -5
40–49 2 44.5 -2 -4
Total N = 25
?
fx
'
= 5
x = 64.5+
(
5
25
×10
)
= 64.5+2 = 66.5
The mean is 66.5, matching the direct method.
Median
The median is the middle value in an ordered dataset, denoted by M
d
. For grouped data, it
assumes linear interpolation within the median class.
UngroupedData(OddNumberofScores)
Formula:
M
d
= valueof the
(
N +1
2
)
th
score
Example 1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30 (N = 9). Ordered: 30, 32, 34, 34, 35,
47, 58, 67, 74.
N +1
2
=
9+1
2
= 5
The 5th score is 35, soM
d
= 35.
Example2: Weights(kg): 60, 65, 70, 55, 68 (N = 5). Ordered: 55, 60, 65, 68, 70.
N +1
2
=
5+1
2
= 3
The 3rd score is 65, soM
d
= 65.
UngroupedData(EvenNumberofScores)
Formula:
M
d
=
valueof
(
N
2
)
th
score +valueof
(
N
2
+1
)
th
score
2
3
Example1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39 (N = 10). Ordered: 30, 32, 34, 34,
35, 39, 47, 58, 67, 74.
M
d
=
35+39
2
= 37
The median is 37.
Example2: Ages: 25, 30, 28, 35, 27, 32 (N = 6). Ordered: 25, 27, 28, 30, 32, 35.
M
d
=
28+30
2
= 29
The median is 29.
GroupedData
Formula:
M
d
=L+
[
N
2
-F
f
m
]
×i
whereListhelowerlimitofthemedianclass,F isthecumulativefrequencybeforethemedian
class,f
m
is the frequencyof the median class, andi isthe class interval.
Example1: DatawithN = 30:
ClassInterval Frequency(f)
35–39 5
30–34 7
25–29 5
20–24 6
15–19 4
10–14 3
Total N = 30
-
N
2
= 15. Cumulative frequencies: 10–14: 3; 15–19: 7; 20–24: 13; 25–29: 18. Median class:
25–29 (L = 24.5,f
m
= 5,F = 13,i = 5).
M
d
= 24.5+
[
15-13
5
]
×5 = 24.5+2 = 26.5
Themedian is 26.5.
Example2: DatawithN = 25:
ClassInterval Frequency(f)
80–89 4
70–79 6
60–69 8
50–59 5
40–49 2
Total N = 25
4
-
N
2
= 12.5. Cumulative frequencies: 40–49: 2; 50–59: 7; 60–69: 15. Median class: 60–69
(L = 59.5,f
m
= 8,F = 7,i = 10).
M
d
= 59.5+
[
12.5-7
8
]
×10 = 59.5+6.875 = 66.375
The median is 66.375.
Mode
The mode is the most frequent score, denoted byMo. For grouped data, it is computed within
the modal class (highest frequency)using interpolation.
UngroupedData
Method: Identify the score(s) with the highest frequency.
Example 1: Scores: 58, 34, 32, 47, 74, 67, 35, 34, 30, 39. The score 34 appears twice, so
Mo = 34.
Example2: Shoesizes: 7, 8, 8, 9, 7, 10, 8. The size 8 appears three times, soMo = 8.
GroupedData
Formula:
Mo =L+
[
d
1
d
1
+d
2
]
×i
whereListhelowerlimitofthemodalclass,d
1
=f
m
-f
m-1
(differencebetweenmodalclass
frequency and the frequency of the class below),d
2
= f
m
-f
m+1
(difference between modal
class frequencyand the frequencyof the class above),andi isthe class interval.
Example1: Usingthe?rstgroupeddata(N = 30): -Modalclass: 30–34(f
m
= 7,L = 29.5).
Below: 25–29(f
m-1
= 5). Above: 35–39(f
m+1
= 5). Thus,d
1
= 7-5 = 2,d
2
= 7-5 = 2,
i = 5.
Mo = 29.5+
[
2
2+2
]
×5 = 29.5+2.5 = 32
The mode is 32.
Example 2: Using the second grouped data (N = 25): - Modal class: 60–69 (f
m
= 8,
L = 59.5). Below: 50–59 (f
m-1
= 5). Above: 70–79 (f
m+1
= 6). Thus, d
1
= 8-5 = 3,
d
2
= 8-6 = 2,i = 10.
Mo = 59.5+
[
3
3+2
]
×10 = 59.5+6 = 65.5
Themode is 65.5.
5