From the population to the sample The sampling distribution FETP India

Competency to be gained from this lecture Use the properties of the sampling distribution to calculate standard error to the mean

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

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 units z = (x-mean)/sd Normal curve  Bell-shaped curve that relates probability to z-scores Parameters and statistics

Population parameters A population parameter is a numerical descriptive measure of a population Examples:  Population mean (µ)  Standard deviation (  ) Parameters and statistics

A statistic A statistic is a numerical descriptive measure of a sample Examples:  Sample mean x  Sample standard deviation s Parameters and statistics

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

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

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

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

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

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

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

Multiple sample means

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

Standard deviation of the sampling distribution If the standard deviation of the population is  The standard deviation of the sampling distribution will be  / (√ n) n is the sample size Sampling distribution

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

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

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 to range is 5 out of 100 Standard error

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

Simple example Let the population be 1,2,3,4,5  Mean = 15/5 = 3 = µ Let’s take a sample of two elements The 25 possible samples are: 1,11,21,31,41,5 2,12,22,32,42,5 3,13,23,33,43,5 4,14,24,34,44,5 5,15,25,35,45,5 Standard error

The frequency distribution of the population is not normal Values Frequency Standard error

Standard deviation of the population Standard error

Looking at the mean of the samples The 25 means of the 25 samples are: Mean of sample means = 75/25 = 3 Same as population mean Standard error

The sampling distribution tends to be normal Values Frequency Even if the population is not normally distributed, the sampling distribution will tend to be normal Standard error

Standard deviation of the sample Standard error

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

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

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

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