Basic statistics

FICCI QUALITY FORUM 1

INTRODUCTION

TO

BASIC STATISTICS

Training on assuring quality of test results


"There are three kinds of lies : lies, damned lies, and statistics".

American humorist Mark TwainOr

British Prime Minister Disraeli



STATISTICS DEALS WITH

1. Collection of data 2. Summarization of data 3. Analysis of data 4. Drawing valid inference from data



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



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



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



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



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



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



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



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.



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



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



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



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



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



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



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



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



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



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



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



NORMAL DISTRIBUTION

1 68.27 % of all observations 2 95.46 % of all observations 3 99.73 % of all observations

EMPERICAL RULE



“I hear and I forget.

I see and I remember.

I do and I understand.”

ConfuciusFamous Chinese

Thinker & Philosopher(551 – 479 BC)


Education

Basic statistics