From the population to the sample

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From the population to the sample

• The sampling distribution FETP India

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Competency to be gained from this lecture

• Use the properties of the sampling distribution
  to calculate standard error to the mean

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Key issues

• Population parameters versus sample statistics
• Sampling distribution and its properties
• Mean and standard error of the sampling
  distribution

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Things we already know

• Mean
• Arithmetic sum of data divided by number of
  observations
• Standard deviation
• Index of variability (spread) of data about the
  mean
• Z-score
• Distance from mean in standard deviation unitsz
  (x-mean)/sd
• Normal curve
• Bell-shaped curve that relates probability to
  z-scores

Parameters and statistics
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Population parameters

• A population parameter is a numerical descriptive
  measure of a population
• Examples
• Population mean (µ)
• Standard deviation (?)

Parameters and statistics
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A statistic

• A statistic is a numerical descriptive measure of
  a sample
• Examples
• Sample mean x
• Sample standard deviation s

Parameters and statistics
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Inference

• The parameter is fixed
• The sample statistics varies from sample to
  sample
• We try to infer what happens in the population
  from what we see in the sample

Parameters and statistics
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Sample mean A typical situation

• A sample might be taken
• The mean and standard deviation are computed
• From this data, one will want to infer that the
  population values are identical or at least
  similar
• In other words, it is hoped that the sample data
  reflects the population data

Sampling distribution
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Sample mean Another approach

• Change your thinking from a single sample
• Consider the situation where you
• Take many samples
• Calculate a mean and standard deviation for each
  sample

Sampling distribution
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Taking many samples from a population

• Consider a population of 1,000 individuals with
  various heights
• If we take 10 samples of 100 persons from the
  population, each of the 10 samples will have a
  specific frequency distribution with
• A specific mean
• A specific standard deviation
• In each sample, each data point is a height

Sampling distribution
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Looking at the means of the samples

• We can look at the frequency distribution of the
  means of each of the 10 samples
• In this case
• The data points are no longer the heights
• The data points are the means

Sampling distribution
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Intuitive observation

• If we take iterative samples from a population,
  we are unlikely to sample extreme values every
  time
• Values close to the mean are common
• Extreme values are less common
• Thus, when we compare the distribution of the
  heights and the distribution of the means, we
  observe
• More variation in the distribution of individual
  heights
• Less variation in the distribution of the means

Sampling distribution
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Taking many samples from the population

• If we take many samples, we can plot a complete
  frequency distribution of the means of the
  samples
• Each sample produces a statistic (mean)
• The distribution of statistics (means) is called
  a sampling distribution

Sampling distribution
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Multiple sample means
Sampling distribution
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Important properties of the sampling distribution

1. The sampling distribution is normally distributed
2. The mean of the sampling distribution is equal to
   the mean of the population

Sampling distribution
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Standard deviation of the sampling distribution

• If the standard deviation of the population is ?
• The standard deviation of the sampling
  distribution will be ? / (v n)
• n is the sample size

Sampling distribution
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Terminology

• The mean of the sampling distribution continues
  to be called the mean
• The standard deviation of the sampling
  distribution is the standard error

Standard error
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Distribution of sample means

• One could obtain a standard deviation of sample
  means which would describe the variability and
  the spread of sample means about the true
  population mean
• In a practical situation
• There is only one sample mean
• One hopes this sample mean is near the real
  population mean
• Wouldn't it be nice to have an estimate of the
  standard deviation of sample means which describe
  the spread of sample means?

Standard error
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Standard error of the mean

• Divide the standard deviation by the square root
  of the number of observations
• The resulting estimate of the standard deviation
  of sample means is called the standard error of
  means
• It can be interpreted in a manner similar to the
  standard deviation of raw scores
• For example, the probability of obtaining a
  sample mean which is outside the -1.96 to 1.96
  range is 5 out of 100

Standard error
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Central limit theorem

• If x possesses any distribution with mean µ and
  standard deviation SD
• Then the sample mean x based on a random sample
  of size n will have a distribution that
  approaches the distribution of a normal random
  variable
• Mean µ
• Standard deviation SD/square root of n as n
  increases without limit.
• Special case
• If x is normally distributed, the result is true
  for any sample size

Standard error
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Simple example

• Let the population be 1,2,3,4,5
• Mean 15/5 3 µ
• Lets take a sample of two elements
• The 25 possible samples are

1,1 1,2 1,3 1,4 1,5 2,1 2,2 2,3 2,4 2,5 3,1 3,2 3,
3 3,4 3,5 4,1 4,2 4,3 4,4 4,5 5,1 5,2 5,3 5,4 5,5
Standard error
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The frequency distribution of the population is
not normal
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Frequency
1
0
1
2
3
4
5
Values
Standard error
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Standard deviation of the population
Standard error
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Looking at the mean of the samples

• The 25 means of the 25 samples are

1 1.5 2 2.5 3 1.5 2 2.5 3 3.5 2 2.5 3 3.5 4 2.5 3
3.5 4 4.5 3 3.5 4 4.5 5
Mean of sample means 75/25 3 Same as
population mean
Standard error
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The sampling distribution tends to be normal
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5
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Frequency
3
2
1
0
1
1.5
2
2.5
3
3.5
4
4.5
5
Values
Even if the population is not normally
distributed, the sampling distribution will tend
to be normal
Standard error
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Standard deviation of the sample
Standard error
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Standard deviation in the population and
standard error

• Standard deviation in the population
• 1.4
• Sample size
• 2
• Square root of the sample size
• 1.4
• Standard deviation / square root of the sample
  size
• 1.4 / 1.4 1
• Standard error

Standard error
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Applying the standard error Male's serum uric
acid levels (1/2)

• Population mean
• 5.4 mg per 100 ml
• Standard deviation is
• 1
• Take 100 samples of 25 men in each sample
• Compute 100 sample means
• How many of those means would you expect to fall
  within the range 5.4-(1.96x1) to 5.4(1.96x1)?
• The answer is 95!

Standard error
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Applying the standard error Male's serum uric
acid levels (2/2)

• One sample
• Mean serum uric acid level of 8.2
• Would you assume this was "significantly"
  different from the population mean?
• Yes, because a mean of that magnitude could occur
  less than 5 times in 100

Standard error
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Key messages

• While population parameters are fixed, samples
  provide estimates (statistics) that fluctuate
• The distribution of a statistic for all possible
  samples of given size n is called the sampling
  distribution.
• For large n, the sampling distribution is
  normal, even if the original distribution is
  not.
• If the original distribution is normal, the
  result is true even for small n.
• The mean of the sampling distribution is the
  population mean and the standard deviation
  (standard error) is the population SD/ sq.root n