14 sampling distribution

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Research Methodology Dr. Nimit Chowdhary, Professor

© Dr. Nimit Chowdhary Research Methodology Workshop p. 2 Saturday, April 14, 2012

There exists a population with its parameters A number of samples can be drawn from this

population Each sample will have its own sample

statistics like sample’s mean standard deviation, etc.

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Distribution of a sample statistic is called a sampling distribution

Sampling distribution is different from sample distribution- distribution of variables within the sample

© Dr. Nimit ChowdharySaturday, April 14, 2012

A baby sitter has five children under her supervision. The average age of these five children is 6 years. However, the age of each child individually is as follows:

Child (X) Age Child (X) Age1 2 4 82 4 5 103 6

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© Dr. Nimit Chowdhary Research Methodology Workshop p. 5

N = 5,5

1

52 4 6 8 10

530 65

ii

X

years


2

5( )

40 85

2.83

N

ii

X

N

years

X2 6 164 6 46 6 08 6 410 6 16

2( )X

2( ) 40X

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Let us take all the samples of size 2 from this population.

There will be 10 samples

1 2 1

1 3 2

1 4 3

1 5 4

2 3 5

2 4 6

1 5 7

3 4 8

3 5 9

4 5 10

, (2, 4) 3

, (2, 6) 4

, (2, 8) 5

, (2,10) 6

, (4, 6) 5

, (4,8) 6

, (4,10) 7

, (6, 8) 7

, (6,10) 8

, (8,10) 9

X X X

X X X

X X X

X X X

X X X

X X X

X X X

X X X

X X X

X X X


10

1

103 4 5 6 5 6 7 7 8 9

1060 610

ii

XX

X

X years

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Sample mean Frequency

Relative Frequency Probability

3 1 1/10 .14 1 1/10 .15 2 2/10 .26 2 2/10 .27 2 2/10 .28 1 1/10 .19 1 1/10 .1

1.0


Sample mean Probability

3 .14 .15 .26 .27 .28 .19 .1

1.0

X ( )P x

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© Dr. Nimit Chowdhary Saturday, April 14, 2012

. ( )(3 .1) (4 .1) (5 .2) (6 .2) (7 .2) (8 .1) (9 .1)6

x

x

x

x P x

years

© Dr. Nimit ChowdharyResearch Methodology Workshop p.

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Sampling distribution of means Sampling distribution of proportions

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© Dr. Nimit Chowdhary Saturday, April 14, 2012

1. Mean of sampling distribution and mean of population is the same

2. Spread of the sample means in the distribution is smaller than the spread in the sample values

3. Sampling distribution of sampling means tend to be bell shaped


Regardless of the shape of the distribution of the population, the distribution of the sample means approaches the normal probability distribution as the sample size increases.

Thus, we can use our knowledge of normal distributions to arrive at conclusions about distribution of sample means

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x

Standard error of mean is the standard deviation (a measure of dispersion) of the distribution of sample means ( ) around mean of sampling distribution.

This mean can be considered as same as population mean

x

© Dr. Nimit ChowdharyResearch Methodology Workshop p.

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2( )xx

xN

N= is the number of samples (not individual sample size)

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3 6 94 6 45 6 16 6 07 6 18 6 49 6 9

X x2( )xX

2( )xx = 28

2( )

28 47

2

xx

x

x

xN

Note: Standard error (2) is smaller than population standard deviation (2.83)

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1x

x

N nNn

n

1N nN

xFor large samples when N>>n

would approach 1

Then,

N = Population size

N = sample size

= populations standard deviation

= standard error of the mean

Note: decreases as sample size increases

x


1N nN

Is the finite correction factor

• Must be used in case of large samples

• Used when sample size is more that 5% of the population size

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The IQ scores of college students are normally distributed with a mean of 120 and standard deviation of 10.

What is the probability that the IQ scores of any one student chosen at random is between 120 and 125

120 = 120 = 10

125

( ) (125 120)10

5 0.510

xZ

Z

Calculating for Z, we get Area for Z=0.5, from Z tables is 0.1915.

Therefore, there are 19.15 % chance that student picked up randomly will have an IQ score between 120 and 125.

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If a random sample of 25 students is taken, what is the probability that the mean of this sample will be between 120 and 125.

We know,

1.This is the distribution of sample means. One of these samples (of size 25) has a mean of 120

2.The mean of the distribution will be same as population mean 120

3.The standard deviation of means around population mean (S.E.) will have to be calculated


. . xS En

10 10 2525x

2.5

( ) 125 1202

5 2.52

0.4938

x

x

z

xZ

Z

Area

2x Therefore,120

125

120x

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Thus there are 49.38% chance that the sample mean would be between 120 and 125.

One can see that as sample size increases, s.d. reduces further and this chance will further increase.

The previous case can be considered as a limiting case when sample size = 1 (the sample size reduces and so does the chance).

…can be defined as a distribution of proportions of all possible random samples of a fixed size n…

p = sample proportion = population proportion

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14 sampling distribution