Unit – 3

Statistics and Probability

3.1 Measures of central tendency

A mean is a value which is representative of a set of data. Average value may also be termed as measures of central tendency. There are five types of averages in common.

(i)            Arithmetic average or mean

(ii)            Median

(iii)            Mode

(iv)            Geometric Mean

(v)            Harmonic Mean

ARITHMETIC MEAN:

(a)  If are n numbers, then their arithmetic mean (A.M.) is defined by

This is known as direct method.

(b)  Short cut method

Let a be the assumed mean, d the deviation of the variate x from a. Then

Example1. Find the arithmetic mean for the following distribution:

Solution: Let assumed mean (a) = 25

(c) Step deviation method

Let be the assumed mean, the width ofthe class interval and

Example 2. Find the arithmetic mean of the data given in example 3 by step deviation method

Solution: Let

MEDIAN:

Median is defined as the measure of the central item when they are arranged in ascending or descending order of magnitude.

When the total number of the items is odd and equal to say , then the value of th item gives the median.

When the total number of the frequencies is even, say , then there are two middle items, and so the mean ofthe values of nth and th items is the median.

Example 3. Find the median of 6, 8, 9, 10, 11, 12, and 13.

Solution: Total number of items

The middle item

Median Value of the 4th item

For grouped data, Median

where is the lower limit of the median class, is the frequency of the class, is the width ofthe class‐interval, is the total ofall the preceding frequencies of the median‐class and is total frequency ofthe data.

Example 4. Find the value of Median from the following data:

Solution:  The given cumulative frequency distribution will first be converted into ordinary frequency as under

Median size of or 327.Item

327. Item lies in 10‐15 which is the median class.

Where stands for lower limit ofmedian class,

Stands for the total frequency,

Stands for the cumulative frequency just preceeding the median class, stands for class interval

Stands for frequency for the median class.

Median

MODE

Mode is defined to be the size of the variable which occurs most frequently.

Example 5: Find the mode of the following items:

.

Solution: 6 occurs 5 times and no other item occurs 5 or more than 5 times, hence the mode is 6.

For grouped data,

where is the lower limit of the modal class, is the frequency of the modal class, is the width of the class, is the frequency before the modal class and is the frequency after the modal class.

Empirical formula

Mean‐ Mode [Mean

Example 6. Find the mode from the following data:

Solution:

Mode

GEOMETRIC MEAN

, , be values of variates , then the geometric mean

Example 7. Find the geometric mean of 4, 8, 16.

Solution: .

HARMONIC MEAN

Harmonic mean of a series of values is defined as the reciprocal of the arithmetic mean of their reciprocals. Thus be the harmonic mean, then

Example 8: Calculate the harmonic mean of 4, 8, 16.

Solution:

3.2 Standard Deviation

It is defined as the positive square root of the arithmetic mean of the square of the deviation of the given values from their arithmetic mean. It is denoted by symbol .

Where is A.M of the distribution. We have more formulae to calculate standard deviation.

….                  

In frequency distribution from, we put

where h is generally taken as width of class interval

1. Calculate S.D for the following distribution.

Solution:

Using formula,

2. Fluctuations in the aggregate of marks obtained by two groups of students are given below.

Solution:

First we represent the data in frequency distribution from group A

For group B,

3.3 Coefficient of Variation:

Coefficient of variation

Formula for calculate Arithmetic Mean (A.M)

3. Calculate coefficient variation for the following frequency distribution.

Solution:

We already calculated

 ….. (refer last Ex.)

Now,   A.M    

A.M    

Coefficient of Variation

4. Refer Question – 2 Calculate the coefficient of variation

Solution:

As we calculate,

σ for Group A σA=11.105

Now A.M

A.M

Coefficient of Variation

Same for Group B,

Now,

Coefficient of Variation

3.4 Moments

The rth moment of a variable x about the mean x is usually denoted by is given by

The rth moment of a variable x aboutany point a is defined by

Relation between moments about mean and moment about any point:

where and

In particular

Note. 1. The sum of the coefficients of the various terms on the right‐hand side is zero.

2. The dimension of each term on right‐hand side is the same as that of terms on the left.

MOMENT GENERATING FUNCTION

The moment generating function of the variate about is defined as the expected value of and is denoted .

where , ‘ is the moment of order about

Hence coefficient of or

again )

Thus the moment generating function about the point moment generating function about the origin.

3.5 Skewness and Kurtosis

Skewness denotes the opposite of symmetry. It is lack of symmetry. In a symmetrical series, the mode, the median, and the arithmetic average are identical.

Coefficient of skewness

KURTOSIS: It measures the degree of peakedness of a distribution and is given by Measure of kurtosis.

Negative skewness      Positive skewness

A.Mesokurtic

B.Leptokurtic

C. Playkurtic

If , the curve is normal or mesokurtic.

If , the curve is peaked or leptokurtic.

If , the curve is flat topped or platykurtic

3.6 Correlation and Regression

Whenever two variables x and y are so related that an increase in the one is accompanied by an increase or decrease in the other, then the variables are said to be correlated.

For example, the yield of crop varies with the amount of rainfall.

If an increase in one variable corresponds to an increase in the other, the correlation is said to be positive. If increase in one corresponds to the decrease in the other the correlation is said to be negative. If there is no relationship between the two variables, they are said to be independent.

Perfect Correlation: If two variables vary in such a way that their ratio is always constant, then the correlation is said to be perfect.

KARL PEARSON’S COEFFICIENT OF CORRELATION:

between two variables x and y is defined by the relation

Where, X = x –, Y = y –

i.e. X, Y are the deviations measured from their respective means,

Example: Ten students got the following percentage of marks in Economics and Statistics

Calculate the co-efficient of correlation.

Solution: Let the marks oftwo subjects be denoted by and respectively.

Then the mean for marks and the mean ofy marks

If and are deviations of x’s and’s from their respective means, then the data may be arranged in the following form:

Here ,  ,

Solution: Let be the ranks of individuals corresponding to two characteristics.

Assuming nor two individuals are equal in either classification, each individual takes the values 1, 2, 3, and hence their arithmetic means are, each

Let , , , be the values of variable and , , those of

Then

where and y are deviations from the mean.

Clearly, and

SPEARMAN’S RANK CORRELATION COEFFICIENT:

where denotes rank coefficient of correlation and refers to the difference ofranks between paired items in two series.

Example: Compute Spearman’s rank correlation coefficient r for the following data:

Solution:

Example: If X and Yare uncorrelated random variables, the of correlation between and

Solution:

Let and

Then

Now

Similarly

Now

Also

(As and are not correlated, we have )

Similarly

REGRESSION

If the scatter diagram indicates some relationship between two variables and, then the dots of the scatter diagram will be concentrated round a curve. This curve is called the curve ofregression. Regression analysis is the method used for estimating the unknown values of one variable corresponding to the known value of another variable.

LINE OF REGRSSION

When the curve is a straight line, it is called a line of regression. A line of regression is the straight line which gives the best fit in the least square sense to the given frequency.

Example: Find the correlation betweenx and , when the lines ofregression are: and

Solution: Let the line of regression ofx on be

Then, the line ofregression ofy on is

and

which is not possible. So our choice of regression line is incorrect.

The regression line ofx on is

And, the regression line ofy on is

And

Hence the correlation coefficient between and is

Example: The following regression equations were obtainedfrom a correlation table:

Find the value of

(a) The correlation coefficient,

(b) The mean and

(c) The mean of

Solution:

(a)  From (1),

(b) From (2),

From (3) and (4)

Coefficient of correlation

(b) (1) and (2) pass through the point .

   (5)

   (6)

On solving (5) and (6), we get

3.7 Reliability of Regression Estimates:

In addition to study the reliability of regression estimates we require to know the standard error.

The standard error of regression estimate of y on

is

2.     The Standard error of Regression estimate of on is

1. Discuss the Reliability of Regression Estimates:

Solution:

For A,

For B,

Now,

The standard error of Regression of estimates of y on x is

…. (Standard error of Regression of estimates of y on x is)

3.8 Probability

Let Aand Bbe two events of a sample space Sand let . Then conditional probability of the event A, given B, denoted byis defined by –

Theorem: If the events Aand Bdefined on a sample space S of a random experiment are independent, then

Example1: A factory has two machines A and B making 60% and 40% respectively of the total production. Machine A produces 3% defective items, and B produces 5% defective items. Find the probability that a given defective part came from A.

Solution: We consider the following events:

A: Selected item comes from A.

B: Selected item comes from B.

D: Selected item is defective.

We are looking for. We know:

Now,

So we need

Since, D is the union of the mutually exclusive events and (the entire sample space is the union of the mutually exclusive events A and B)

Example2: Two fair dice are rolled, 1 red and 1 blue. The Sample Space is

S = {(1, 1),(1, 2), . . . ,(1, 6), . . . ,(6, 6)}.Total -36 outcomes, all equally likely (here (2, 3) denotes the outcome where the red die show 2 and the blue one shows 3).

(a) Consider the following events:

A: Red die shows 6.

B: Blue die shows 6.

Find, and .

Solution:

NOTE: so for this example. This is not surprising - we expect A to occur in of cases. In of these cases i.e. in of all cases, we expect B to also occur.

(b) Consider the following events:

C: Total Score is 10.

D: Red die shows an even number.

Find , and .

Solution:

NOTE: so,.

Why does multiplication not apply here as in part (a)?

Answer: Suppose C occurs: so the outcome is either (4, 6), (5, 5) or (6, 4). In two of these cases, namely (4, 6) and (6, 4), the event D also occurs. Thus

Although , the probability that D occurs given that C occurs is .

We write, and call the conditional probability of D given C.

NOTE: In the above example

Example3: Three urns contain 6 red, 4 black; 4 red, 6 black; 5 red, 5 black balls respectively. One of the urns is selected at random and a ball is drawn from it. If the ball drawn is red find the probability that it is drawn from the first urn.

Solution:

:The ball is drawn from urnI.

: The ball is drawn from urnII.

: The ball is drawn from urnIII.

R:The ball is red.

We have to find

Since the three urns are equally likely to be selected

Also,

From (i), we have

3.9 Probability Density function

If be any continuous random variable then the p.d.f then a function is said to be the P.D.F if

Here, Random Variable

Event

P.D.F

P.M.F (discrete variable)

Constant density function

Properties

for all

1. If

Find (i)    (ii)      (iii) 

Solution:

(i) Condition of P.D.F

(ii)

(iii)

3.10 Probability distributions: Binomial

The Normal Distribution:

The normal distribution is sometimes informally called the bell curve.

The probability density of the normal distribution is:

   is mean or expectation of the distribution

 is the variance

Properties of a normal distribution:

Binomial Distribution:

A distribution is said to be binomial distribution if the following conditions are met.

If all the above conditions met then the binomial distribution describes the probability of X successes in n trials.

A classic example of the binomial distribution is the number of heads (X) in n coin tosses.

The Notation for a binomial distribution is

X ~ B (n, π)

which is read as ‘X is distributed binomial with n trials and probability of success in one trial equal to π ’.

Using this formula, the probability distribution of a binomial random variable X can be calculated if n and π are known.

n! is called ‘n factorial’ = n(n-1)(n-2) . . .(1)

P(X) = #of Scenario * Single Scenario

The first factorial terms gives the number of scenario and the second term describes the probability of success to power of number of successes and probability of failure to the power of number of failures.

3.11 Poisson

Another probability distribution for discrete variables is the Poisson distribution. The Poisson distribution is used to determine the probability of the number of events occurring over a specified time or space. This was named for Simeon D. Poisson, 1781 – 1840, French mathematician.

Examples of events over space or time: -number of cells in a specified volume of fluid

-number of calls/hour to a help line

-number of emergency room beds filled/ 24 hours

Like the binomial distribution and the normal distribution, there are many Poisson distributions.

The probability that there are exactly X occurrences in the specified space or time is equal to

The horizontal axis is the index X. The function is defined only at integer values of X. The connecting lines are only guides for the eye and do not indicate continuity. Notice that as λ increases the distribution begins to resemble a normal distribution.

3.12 Normal

Probability density function or p.d.f. specified the probability per unit of the random variable. Here is an example of a p.d.f. of the daily waiting time by the taxi driver of Uber Taxi Company. In the X axis, daily waiting time and Y-axis probability per hour has been shown.

If one Uber taxi driver want to know the probability to wait more than 7 hours in a day? Then he will be interested in the yellow surface arear shown above. On basis of this graph you can estimate the area. Same thing you can get form below

Cumulative probability curve.

.

Probability to wait more than 7 hours will be calculated using complementary rule 1- P. Because corresponding to 7 in X axis we marked the probability is P and we are interested in more than 7 hours. So, P should be subtracted from 1 to get desired result.

If distribution is bell shape then it is assumed that about 68% of the elements have a z-score between -1 and 1; about 95% have a z-score between -2 and 2; and about 99% have a z-score between -3 and 3.

Assume the time you spend in week days by traveling has given by a normal distribution with mean= 40 mins and SD= 10 mins.

What will be your range of travel time for 95 % of your week days?

As you know 95 % will come within 2 standard deviation of your mean. So, the range will be (40-20) = 20 to (40+20) =60 mins.

Now another question you want to answer that what will be the probability to be travelling more than 50 mins?

Actually you are interested in the yellow surface given in above diagram. You know that a normal distribution is symmetric. So, half of the probability located one side of the mean and another half located another side of the mean.

As SD =10. So, one standard deviation will be 30 to 50 range.

You already know for left side up 40 the probability is 0.5. Now if you calculate the probability from 40 to 50 range it will be half of 1 Standard deviation i.e.    0.68/2 = 0.34

So the probability to travel less than 50 mins = 0.5 +. 0.34 = 0.84

But you are interested in more than 50 mins traveling time so it will be 1- 0.84 =0.16

3.13 Test of Hypothesis: Chi-Square test and t – test:

Chi – Square Test

Observed Frequency

Expected (Theoretical) Frequency

Then Chi – Square distribution is defined as

Where, Total Frequency

Chi-Square () Test: Working Rule

Step I:

Consider the null hypothesis () and Alternative Hypothesis ()

Step II:

Consider expected frequency corresponding to each cell.

Step III:

Calculate distribution by the formula

and calculate the degree of freedom(.

Step IV:

See the value of from the table and the value calculate in step (3).

Step V:

If Calculated value of <Tabulated value of

Then Null hypothesis is accepted otherwise accepted.

Calculation of Expected frequency 2×4 Table

Expected frequencies of cell (1,1) is given by

Calculation of Degree of freedom ():

Binomial Distribution 

Poisson’s Distribution

Normal Distribution

1. In a sample survey of public opinion answer to the question

(i)            Do you drink?

(ii)            Are you in favour of local option on sale of liquor?

Are tabulate below:

Can you infer or not the local option on the safe of liquor is depend on individual drink?

(Given that the value of for dof at 5% level of significance is 3.841)

Solution:

Step 1:

Null Hypothesis . The option on the safe of liquor is not depend with the individual drinking.

Step 2:

Calculation of expected frequency (Theoretical Frequency)

Step 3:

Calculation of distribution. We know that

   (1)

Decision:

Clearly calculated value of

Tabulated value of (3.841)

Null hypothesis is accepted.

Safe of liquor is not depending with the individual drink.

T – Test:

(i)            T – Test is a small sample test

(ii)            It is also called student T – Test.

(iii)    Also called as Welch t – Test.

Uses of T – Test:

Size of sample is small

.

Degree of freedom is

.

Reference Books