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
Measures of Central Tendency
Characteristics of a Measure of Central Tendency
Single number that represents the entire set of data (average)
Three Measures of Central Tendency
The Mean
• The sum of the scores divided by the number of scores
Formula for finding the Mean
• Symbolized by M or “X-bar”
N
XM
Characteristics of the Mean
• The mean may not necessarily be an actual score in a distribution
Example of Finding the Mean
• X: 8, 6, 7, 11, 3
• Sum = 35
• N = 5
• M = 7
The Median
• The point in a distribution that divides it into two equal halves
• Symbolized by Md
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
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
The Mode
• Score or qualitative category that occurs with the greatest frequency
• Always used with nominal data, we find the most frequently occurring category
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
GROUPED DATA
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
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.
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.
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=
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.
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.
MODE
Mode = (3median – 2 mean)
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.
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.