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Basic Stats - Courtesy FICCI
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FICCI QUALITY FORUM 1
INTRODUCTION
TO
BASIC STATISTICS
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"There are three kinds of lies : lies, damned lies, and statistics".
American humorist Mark TwainOr
British Prime Minister Disraeli
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STATISTICS DEALS WITH
1. Collection of data 2. Summarization of data 3. Analysis of data 4. Drawing valid inference from data
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SAMPLE
A product or quantity of a product chosen at random to represent all products in a batch
Sample measurements would mean a finite number of measurements out of a possibility of large number of measurements
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POPULATION
When an infinite number of observations are made in a measurement process, it would result in infinite possible outcomes.
With each possible outcome, we may associate a sample point.
Totality of these sample points is called statistical universe or population
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DATA SUMMARIZATION METHODS ARE
1. GRAPHICAL
Bar Chart, Pie Chart etc.
2. TUBULAR
Histogram
3. NUMERICA L
a) Measures of central tendency b) Measures of dispersion
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Important measures of central tendency are :
1. Mean2. Median3. Mode
Important measures of dispersion or variability of a set of numbers or data are :
1. Range2. Deviation3. Variance4. Standard deviation
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ARITHMATIC MEAN OR AVERAGE
The sum of the results divided by the number of measurements.
If x1, x2, x3, .... xn, are the results of measurements, then the average is given by
= 1/n (x1+ x2 + x3 + .... + xn)Example - 1
Data = 45, 48, 35, 39, 51, 42, 45, 39, 45, 44, 45
Average = 43.45
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MODE :It is the value which occurs the maximum in the set of data
Example of computing Mode
Example - 2Data 25, 23, 22, 31, 35, 32, 30, 33, 22Arranged data 22, 22, 23, 25, 30, 31,
32, 33, 35Mode = 22
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Example - 3
Data 45, 48, 35, 39, 51, 42, 39, 45, 44
Arranged data 35, 39, 39, 42, 44, 45,45, 48, 51
Mode = 39 & 45
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MODE :
The mode can be very useful for dealing with categorical data.
For example, if a PIZZA shop sells 10 different types of PIZZA, the mode would represent the most popular PIZZA.
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MEDIAN :It is the middle value of a set of numbers arranged according to magnitude. If two middle values exist in an even number of determinations, the median is the mean of the two middle values.
Advantage of Median over Mean is that it is not affected by extreme values
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To compute Median
i) Arrange data from smallest to largestii) For odd No. of data,
Median = central valueiii) For even No. of data,
Median = Av. of 2 central values
Example - 4Data 25, 23, 22, 31, 35, 32, 30, 33Arranged data 22, 23, 25, 30, 31, 32, 33, 35Median = (30 + 31) / 2 = 30.5
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Example - 5
Data 45, 48, 35, 39, 51, 42, 45, 39, 45, 44, 45
Arranged data 35, 39, 39, 42, 44, 45, 45, 45, 45, 48, 51
Median = 45
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RANGE
It is the difference between the largest and the smallest value in a set of data
Examples-6
DATA RANGE
16.5, 11.2, 10.4, 14.1, 17.2, 15.8 6.8
4.52, 4.81, 3.55, 3.97, 5.11, 4.28, 1.56
22, 23, 25, 31, 35, 32, 30 13
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It is the difference between the measured value of a quantity and a reference value
If is the reference value, the deviation, is given by = (x1 )
The mean deviation, m = 1/n[(x1 ) + (x2 )+ ...+ (xn )]
Where ‘n’ = No. of observations
DEVIATION
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VARIANCE
It is the average of the squared differences between the individual measurements and the mean
If 2 is the variance, then
2 = 1/n[(x1 )2 + (x2 )2+ ...+ (xn )2]
It is defined as the square of the standard deviation
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It is defined as the root-mean-square deviation of the observed numbers from their arithmetic mean. It is the positive square root of the variance
Population Standard Deviation _____________________________
n = 1/n * [(X1- )2+(X2- )2+…..+(Xn- )2]
Where is the mean value of “n” observations
STANDARD DEVIATION
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When No. of observations, ‘n’ is less than 25, Standard Deviation is calculated from the formula
Sample Standard Deviation ________________________________
n-1 = 1/(n-1) * [(X1- )2+(X2- )2+…..+(Xn- )2]
Where is the mean value of “n” observations
SAMPLE STANDARD DEVIATION
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EVALUATION OF STANDARD DEVIATION
EXAMPLE - 7
DATA100.2, 100.5, 100.6, 100.4, 100.3, 100.2, 100.5, 100.6, 100.5, 100.3
Average = 100.41
Standard Deviation, n-1 = 0.152
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No DATA (Xn) DEVIATION (Xav – Xn)VARIANCE(Xav – Xn)2
1 100.2 100.41-100.2 = 0.21 0.04412 100.5 100.41-100.5 = -0.09 0.00813 100.6 100.41-100.6 = -0.19 0.03614 100.4 100.41-100.4 = 0.01 0.00015 100.3 100.41-100.3 = 0.11 0.01216 100.2 100.41-100.2 = 0.21 0.04417 100.5 100.41-100.5 = -0.09 0.00818 100.6 100.41-100.6 = -0.19 0.03619 100.5 100.41-100.5 = -0.09 0.008110 100.3 100.41-100.3 = 0.11 0.0121
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Add the Variances, i.e.
(Xav – Xn)2 = 0.2090
Divide the above by (n – 1) = 10 – 1 = 9
i.e. 0.2090 / 9 = 0.0232
Square root of this value gives us the Standard Deviation, i.e.
n-1 = 0.0232 = 0.152
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NORMAL DISTRIBUTION
1 68.27 % of all observations 2 95.46 % of all observations 3 99.73 % of all observations
EMPERICAL RULE
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“I hear and I forget.
I see and I remember.
I do and I understand.”
ConfuciusFamous Chinese
Thinker & Philosopher(551 – 479 BC)
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