Mean, median, and mode ug

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Transcript of Mean, median, and mode ug
Single value in series of observations which indicate the characteristics of observations
All data / values clustered around it & used to compare between one series to another
Measures: a) Mean (Arithmetic / Geometric / Harmonic)
b) Medianc) Mode
It is sum of all observations divided by number of observations __ Σx Mean ( X ) =  ( x= observation & n= no of observations) n
Problem: ESR of seven subjects is 8,7, 9, 10, 7, 7 and 6. Calculate the mean. 8+7+9+10+7+7+6 54 Mean=  =  = 7.7 7 7
For discrete observation:
If we have x1, x2, …… xn observations with corresponding frequencies f1, f2, ……fn, then
x1 f1+ x2 f2+ ……. xn fn Σfx Mean =  =  f1+ f2+ ……fn Σf Problem: Calculate the avg. no. of children / family from the
following data:
No. of Children (X) No. of families ( f ) Total no of children (fx)
0 30 0 x 30 = 01 52 1 x 52 = 522 60 2 x 60 = 1203 65 3 x 65 = 1954 18 4 x 18 = 725 10 5 x 10 = 506 5 6 x 5 = 30
Total = 240 = 519
Mean = 519/ 240 = 2.163
When observations are arranged in ascending or descending order of magnitude, the middle most value is known as Median
Problem: Same example of ESR as in mean observations are arranged first in ascending order, i.e 6, 7, 7, 7, 8, 9, 10
n+1 7+1 When n is odd, Median =  th observation i.e,  = 4th observation = 7 2 2 n/2 th + (n/2 +1) th observation When n is even, Median =  2 So, if there are 8 observations of ESR like 5, 6, 7, 7,7, 8, 9, 10 n/2 th + (n/2 +1) 4th + 5th 7+ 7 Median = th observation = th observation =  = 7
= 2 2 2
The mode is the data item that appears the most.
If all data items appear the same number of times, then there is no mode.
5, 4, 6, 11, 5, 7, 10, 5
The mode is 5.
a. 5 5 5 3 1 5 1 4 3 5
b. 1 2 2 2 3 4 5 6 6 6 7 9
c. 1 2 3 6 7 8 9 10
Examples
Mode is 5
Bimodal  2 and 6
No Mode
Merits DemeritsMean:
• Rigidly defined• Based on all observations• Easy to calculate & understand• Least affected by sampling fluctuation, hence more stable
Mean:
• Can be used only for quantitative data• Unduly affected by extreme observations
Median:
• Not affected by extreme observations• Both for quantitative & qualitative data
Median:
• Affected more by sampling fluctuations• Not rigidly defined • Can be used for further mathematical calculation
Mode:
• Not affected by extreme observations• Both for quantitative & qualitative data
Mode:
• Not rigidly defined • Can be used for further mathematical calculation
SymmetricData is symmetric if the left half of its
histogram is roughly a mirror of its right half.
SkewedData is skewed if it is not symmetric
and if it extends more to one side than the other.
Definitions
Skewness
Mode = Mean = Median
SYMMETRIC
Figure 213 (b)
Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
Figure 213 (b)
Figure 213 (a)
Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
SKEWED RIGHT(positively)
Mean Mode Median
Figure 213 (b)
Figure 213 (a)
Figure 213 (c)
Biological variation in large groups is common. e.g : BP, wt
What is normal variation? and How to measure?
Measure of dispersion helps to find how individual observations are dispersed around the central tendency of a large series
Deviation = Observation  Mean
Range
Quartile deviation
Mean deviation
Standard deviation
Variance
Coefficient of variance : indicates relative variability (SD/Mean) x100
Range : difference between the highest and the lowest value
Problem: Systolic and diastolic pressure of 10 medical students are as follows:
140/70, 120/88, 160/90, 140/80, 110/70, 90/60, 124/64, 100/62, 110/70 & 154/90. Find out the range of systolic and diastolic blood pressure
Solution: Range of systolic blood pressure of medical students: 90160 or 70 Range of diastolic blood pressure of medical students: 6090 or 30
Mean Deviation: average deviations of observations from mean value _ Σ (X – X ) __ Mean deviation (M.D) =  , ( where X = observation, X = Mean n n= number of observation )
Problem: Find out the mean deviation of incubation period of measles of 7 children, which are as follows: 10, 9, 11, 7, 8, 9, 9.
Solution:
Observation (X)
__Mean ( X )
__Deviation (X  X)
10 __
X = Σ X / n = 63 / 7 = 9
1
9 0
11 2
7 2
8 1
9 0
9 0
ΣX=63 _Σ (XX) = 6, ignoring + or  signs
Mean deviation (MD) = _ Σ X  X =  n
= 6 / 7 = 0.85
It is the most frequently used measure of dispersion
S.D is the RootMeansSquareDeviation
S.D is denoted by σ or S.D ___________ Σ ( X – X ) 2 S.D (σ) = γ n
Calculate the mean ↓ Calculate difference between each observation and mean ↓ Square the differences ↓ Sum the squared values ↓ Divide the sum of squares by the no. observations (n) to get ‘mean square
deviation’ or variances (σ2). [For sample size < 30, it will be divided by (n1)]
↓ Find the square root of variance to get RootMeansSquareDeviation or S.D
(σ)
Observation (X)
__Mean ( X )
_Deviation (X X)
__
(XX) 2
58 __ X = Σ X / n = 984/12 = 82
12 576
66 16 256
70 12 144
74 8 64
80 2 4
86 4 16
90 8 64
100 18 324
79 3 9
96 14 196
88 6 36
97 15 225
Σ X = 984 _ Σ (X  X)2 =1914
S.D (σ ) = = Σ(X –X) 2 / n1
=(√1924/ (121) _____= √174
= 13.2
Estimation of Standard DeviationRange Rule of Thumb
x  2s x x + 2s
Range 4sor
(minimumusual value)
(maximum usual value)
Estimation of Standard DeviationRange Rule of Thumb
x  2s x x + 2s
Range 4sor
(minimumusual value)
(maximum usual value)
Range
4s
Estimation of Standard DeviationRange Rule of Thumb
x  2s x x + 2s
Range 4sor
(minimumusual value)
(maximum usual value)
Range
4s =
highest value  lowest value
4
minimum ‘usual’ value (mean)  2 (standard deviation)
minimum x  2(s)
minimum ‘usual’ value (mean)  2 (standard deviation)
minimum x  2(s)
maximum ‘usual’ value (mean) + 2 (standard deviation)
maximum x + 2(s)
x
The Empirical Rule(applies to bellshaped distributions)FIGURE 215
x  s x x + s
68% within1 standard deviation
34% 34%
The Empirical Rule(applies to bellshaped distributions)FIGURE 215
x  2s x  s x x + 2sx + s
68% within1 standard deviation
34% 34%
95% within 2 standard deviations
The Empirical Rule(applies to bellshaped distributions)
13.5% 13.5%
FIGURE 215
x  3s x  2s x  s x x + 2s x + 3sx + s
68% within1 standard deviation
34% 34%
95% within 2 standard deviations
99.7% of data are within 3 standard deviations of the mean
The Empirical Rule(applies to bellshaped distributions)
0.1% 0.1%
2.4% 2.4%
13.5% 13.5%
FIGURE 215