thiyagu measures of central tendency final
DESCRIPTION
THIS SLIDES EXPLORES THE CONCEPT OF MEASURES OF CENTRAL TENDENCY. AND ALSO GIVE THE DEFINITION WITH EXAMPLE OF MEAN, MEDIAN AND MODE.TRANSCRIPT
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Measures of Central Tendency
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Characteristics of a Measure of Central Tendency
Single number that represents the entire set of data (average)
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Three Measures of Central Tendency
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The Mean
• The sum of the scores divided by the number of scores
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Formula for finding the Mean
• Symbolized by M or “X-bar”
N
XM
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Characteristics of the Mean
• The mean may not necessarily be an actual score in a distribution
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Example of Finding the Mean
• X: 8, 6, 7, 11, 3
• Sum = 35
• N = 5
• M = 7
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The Median
• The point in a distribution that divides it into two equal halves
• Symbolized by Md
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Finding the Median
1. Arrange the scores in ascending or descending numerical order
2. Calculate the value of (N+1/2)
3. round the (N+1/2)th item
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Example of Finding the Median
• X: 6, 6, 7, 8, 9, 10, 11
• Median = 8
• Y: 1, 3, 5, 6, 8, 12
• Median = 5.5
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The Mode
• Score or qualitative category that occurs with the greatest frequency
• Always used with nominal data, we find the most frequently occurring category
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Example of Finding the Mode
• X: 8, 6, 7, 9, 10, 6
• Mode = 6
• Y: 1, 8, 12, 3, 8, 5, 6
• Mode = 8
• Can have more than one mode
• 1, 2, 2, 8, 10, 5, 5, 6
• Mode = 2 and 5
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GROUPED DATA
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Mean
Midpoint x CI f fX
95.5 91-100 5 477.5
85.5 81-90 10 855
75.5 71-80 15 1132.5
65.5 61-70 10 655
55.5 51-60 6 333
45.5 41-50 3 136.5
35.5 31-40 1 35.5
N = 50 fX =3625
N
fXMean
M = 3625/50 = 72.5
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Merits of Arithmetic Mean
• (1) Simple to understand
• (2) Easy to compute,
• (3) Capable of further mathematical treatment,
• (4) Calculated on the basis of all the items of the series,
• (5) It gives the value which balances the either side,
• (6) Can be calculated even if some values of the series are missing.
• (7) It is least affected by fluctuations in sampling.
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Demerits of Arithmetic Mean
• (1) Extreme items have disproportionate effect.
• (2) When data is vast, the calculations become tedious.
• (3) In case of open end classes, mean can only be calculated by making some assumptions.
• (4) A. M. is not representative if series is asymmetrical.
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MEDIAN
Exact limit CI f cf
55.5-60.5 56-60 6 60
50.5-55.5 51-55 9 54
45.5-50.5 40-50 15 45
40.5 (L)-45.5 41-45 13 (f) 30
35.5-40.5 36-40 10 17 (M)
30.5-35.5 31-35 7 7
N = 60
302
60
513
)17260(5.40
5.4115.4013
135.405
13
17305.40
cf
mNL
)2(
LOCATION OF THE MEDIAN CLASS
MEDIAN=
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Merits of Median
• (1) Easy to calculate,
• (2) Can be calculated even if the data is incomplete,
• (3) It is unaffected in case of asymmetrical series,
• (4) Useful in case the series of qualitative characteristics is given for example beauty, intelligence etc.
• (5) Median is a reliable measure of central tendency if in a series, frequencies do not tend to be evenly distributed.
• (6) Median can be expressed graphically.
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Demerits of Median
• (1) Calculation of median requires arraying of items which may be tedious if the data is large,
• (2) It is not suitable for further arithmetic treatment because its value is only positional and not mathematical,
• (3) Affected by number of items and not values,
• (4) It is very unstable. In case of any addition to the series, the value of median would change,
• (5) Items of extremes are given no importance.
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MODE
Mode = (3median – 2 mean)
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Merits of mode
• (1) Easy to understand,
• (2) Simple to calculate and locate,
• (3) Quantitative data in ranking is possible, mode is very useful
• (4) It is the actual value that is in the series,
• (5) Mode remains unaffected by dispersion of series,
• (6) Not affected by extreme items,
• (7) Can be calculated even if extreme values are not known.
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Demerits of Mode
• (1) Mode cannot be subject to further Mathematical treatment, because is not obtained from any algebraic calculations.
• (2) It is quite likely that there is no mode for a series,
• (3) Cannot be used if relative, importance of items have to be considered,
• (4) Choice of grouping has a considerable influence on the value of the mode.