Mean, median, and mode ug
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Transcript of Mean, median, and mode ug
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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
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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
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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
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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
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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.
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5, 4, 6, 11, 5, 7, 10, 5
The mode is 5.
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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
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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
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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
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Skewness
Mode = Mean = Median
SYMMETRIC
Figure 2-13 (b)
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Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
Figure 2-13 (b)
Figure 2-13 (a)
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Skewness
Mode = Mean = Median
SKEWED LEFT(negatively)
SYMMETRIC
Mean Mode Median
SKEWED RIGHT(positively)
Mean Mode Median
Figure 2-13 (b)
Figure 2-13 (a)
Figure 2-13 (c)
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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
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Range
Quartile deviation
Mean deviation
Standard deviation
Variance
Coefficient of variance : indicates relative variability (SD/Mean) x100
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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: 90-160 or 70 Range of diastolic blood pressure of medical students: 60-90 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 )
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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 (X-X) = 6, ignoring + or - signs
Mean deviation (MD) = _ Σ X - X = ------------ n
= 6 / 7 = 0.85
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It is the most frequently used measure of dispersion
S.D is the Root-Means-Square-Deviation
S.D is denoted by σ or S.D ___________ Σ ( X – X ) 2 S.D (σ) = γ---------------------- n
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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 (n-1)]
↓ Find the square root of variance to get Root-Means-Square-Deviation or S.D
(σ)
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Observation (X)
__Mean ( X )
_Deviation (X- X)
__
(X-X) 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 / n-1
=(√1924/ (12-1) _____= √174
= 13.2
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Estimation of Standard DeviationRange Rule of Thumb
x - 2s x x + 2s
Range 4sor
(minimumusual value)
(maximum usual value)
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Estimation of Standard DeviationRange Rule of Thumb
x - 2s x x + 2s
Range 4sor
(minimumusual value)
(maximum usual value)
Range
4s
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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
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minimum ‘usual’ value (mean) - 2 (standard deviation)
minimum x - 2(s)
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minimum ‘usual’ value (mean) - 2 (standard deviation)
minimum x - 2(s)
maximum ‘usual’ value (mean) + 2 (standard deviation)
maximum x + 2(s)
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x
The Empirical Rule(applies to bell-shaped distributions)FIGURE 2-15
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x - s x x + s
68% within1 standard deviation
34% 34%
The Empirical Rule(applies to bell-shaped distributions)FIGURE 2-15
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x - 2s x - s x x + 2sx + s
68% within1 standard deviation
34% 34%
95% within 2 standard deviations
The Empirical Rule(applies to bell-shaped distributions)
13.5% 13.5%
FIGURE 2-15
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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 bell-shaped distributions)
0.1% 0.1%
2.4% 2.4%
13.5% 13.5%
FIGURE 2-15