# Population and Sampling Distribution

**Topics:**Sample size, Normal distribution, Standard deviation

**Pages:**21 (2936 words)

**Published:**May 11, 2013

|6.1 POPULATION AND SAMPLING DISTRIBUTION |

|6.1.1 Population Distribution |

Suppose there are only five students in an advanced statistics class and the midterm scores of these five students are: 7078808095

Let x denote the score of a student.

• Mean for Population

Based on Example 1, to calculate mean for population:

[pic]

• Standard Deviation for Population

Based on example 1, to calculate standard deviation for population:

[pic]

|6.1.2 Sampling Distribution |

▪ Sample statistic such as median, mode, mean and standard deviation

6.1.2.1 The Sampling Distribution of the Sample Mean

Reconsider the population of midterm scores of five students given in example 1. Let say we draw all possible samples of three numbers each and compute the mean.

Total number of samples = 5C3 =[pic]

Suppose we assign the letters A, B, C, D and E to scores of the five students, so that

A = 70,B = 78,C=80,D = 80, E = 95

Then the 10 possible samples of three scores each are

ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE

• Sampling Error

▪ Sampling error is the difference between the value of a sample statistic and the value of the corresponding population parameter. In the case of mean,

Sampling error =[pic]

• Mean ( ) for the Sampling Distribution of[pic]

Based on example 2,

[pic]

Standard Deviation ( ) for the Sampling Distribution of

[pic]

Where: [pic] is the standard deviation of the population n is the sample size

This formula is used when

When N is the population size

[pic]

Where is the finite population correction factor

This formula is used when[pic]

▪ The spread of the sampling distribution of [pic] is smaller than the spread of the corresponding population distribution, [pic]. ▪ The standard deviation of the sampling distribution of [pic]decreases as the sample size increase. ▪ The standard deviation of the sample means is called the standard error of the mean.

|6.1.3 Sampling From a Normally Distributed Population |

The criteria for sampling from a normally distributed population are: • [pic]

• [pic]

• The shape of the sampling distribution of [pic]is normal, whatever the value of n. o Shape of the sampling distribution

|6.1.4 Sampling From a Not Normally Distributed Population |

• Most of the time the population from which the samples are selected is not normally distributed. In such cases, the shape of the sampling distribution of [pic] is inferred from central limit theorem.

Central limit theorem

▪ For a large sample size, the sampling distribution of [pic] is approximately normal, irrespective of the...

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