How to Estimate a Mean or Proportion from a Stratified Sample

This lesson describes how to estimate a population mean or proportion, given survey data from a stratified random sample. A good analysis should provide two outputs:

• A point estimate of the population mean or proportion.
• A quantitative measure of uncertainty associated with the point estimate (e.g., a margin of error and/or a confidence interval).

First, we describe how to conduct a good analysis step-by-step. Then, we will illustrate the analysis with a sample problem.

How to Analyze Survey Data

Any good analysis of survey data from a stratified sample includes the same seven steps:

• Estimate a population parameter.
• Compute sample variance within each stratum.
• Compute standard error.
• Specify a confidence level.
• Find the critical value (often a z-score or a t-score).
• Compute margin of error.
• Define confidence interval.

Let's look a little bit closer at each step - what we do in each step and why we do it. When you understand what is really going on, it will be easier for you to apply formulas correctly and to interpret analytical findings.

Note: The formulas presented below are only appropriate for stratified random sampling.

Estimating a Population Mean or Proportion

The first step in the analysis is to develop a point estimate for the population mean or proportion. Use this formula to estimate the population mean:

Sample mean = x = Σ( N_h / N ) * x_h

where N_h is the number of observations in stratum h of the population, N is the number of observations in the population, and x_h is the mean score from the sample in stratum h.

A proportion is a special case of the mean. Use this formula to estimate the population proportion:

Sample proportion = p = Σ( N_h / N ) * p_h

where N_h is the number of observations in stratum h of the population, N is the number of observations in the population, and p_h is the sample proportion in stratum h.

Note that each of the above equations includes the sampling fraction, N_h/N. Thus, to compute a sample estimate of the population mean or population proportion, we need to know the sampling fraction (i.e., we need to know the relative population size of each stratum).

Also, because different samples can produce different point estimates, you can be fairly sure that the estimate from your sample does not equal the true value of the population parameter exactly. Therefore, you need a way to express the uncertainty inherent in your estimate. The remaining six steps in the analysis are geared toward quantifying the uncertainty in your estimate.

Computing Variance Within Strata

The variance is a numerical value used to measure the variability of observations in a group. If individual observations vary greatly from the group mean, the variance is big; and vice versa.

Given a stratified random sample, we need to compute the sample variance within each stratum (s²_h):

s²_h = Σ ( x_ih - x_h )² / ( n_h - 1 )

where s²_h is a sample estimate of population variance in stratum h, x_ih is the value of the ith element from stratum h, x_h is the sample mean from stratum h, and n_h is the number of sample observations from stratum h.

With a proportion, the variance within each stratum can be estimated from a sample as:

s²_h = [ n_h / (n_h - 1) ] * p_h * (1 - p_h)

where s²_h is a sample estimate of the variance within stratum h, n_h is the number of observations from stratum h in the sample, and p_h is a sample estimate of the proportion is stratum h.

Why do we care about the variance within each stratum? Stratum variance is needed to compute the standard error. And why do we care about the standard error? Read on.

Computing Standard Error

The standard error is possibly the most important output from our analysis. It allows us to compute the margin of error and the confidence interval.

When we estimate a mean or a proportion from a stratified random sample, the standard error (SE) of the estimate is:

SE = (1 / N) * sqrt { Σ [ N²_h * ( 1 - n_h/N_h ) * s²_h / n_h ] }

where n_h is the number of sample observations from stratum h, N_h is the number of elements from stratum h in the population, N is the number of elements in the population, and s²_h is a sample estimate of the population variance in stratum h.

Think of the standard error as the standard deviation of a sample statistic. In survey sampling, there are usually many different subsets of the population that we might choose for analysis. Each different sample might produce a different estimate of the value of a population parameter. The standard error provides a quantitative measure of the variability of those estimates.

Specifying Confidence Level

In survey sampling, different samples can be randomly selected from the same population; and each sample can often produce a different confidence interval. Some confidence intervals include the true population parameter; others do not.

A confidence level refers to the percentage of all possible samples that produce confidence intervals that include the true population parameter. For example, suppose all possible samples were selected from the same population, and a confidence interval were computed for each sample. A 95% confidence level implies that 95% of the confidence intervals would include the true population parameter.

As part of the analysis, survey researchers choose a confidence level. Probably, the most frequently chosen confidence level is 95%.

Finding Critical Value

Often expressed as a t-score or a z-score, the critical value is a factor used to compute the margin of error. To find the critical value, follow these steps:

• Compute alpha (α): α = 1 - (confidence level / 100)
• Find the critical probability (p*): p* = 1 - α/2
• To express the critical value as a z-score, find the z-score having a cumulative probability equal to the critical probability (p*).
• To express the critical value as a t-score, follow these steps:
  • Find the degrees of freedom (df). To compute degrees of freedom for a stratified random sample, use this equation:

    df = Σ ( n_h - 1 )

    where n_h is the number of sample observations from stratum h.

  • The critical t-score is the t statistic having degrees of freedom equal to df and a cumulative probability equal to the critical probability (p*).

Researchers use a t-score when sample size is small; a z-score when it is large (at least 30). You can use the Normal Distribution Calculator to find the critical z-score, and the t Distribution Calculator to find the critical t statistic.

Computing Margin of Error

The margin of error expresses the maximum expected difference between the true population parameter and a sample estimate of that parameter.

Here is the formula for computing margin of error (ME):

ME = SE * CV

where SE is standard error, and CV is the critical value.

Defining Confidence Interval

Statisticians use a confidence interval to express the degree of uncertainty associated with a sample statistic. A confidence interval is an interval estimate combined with a probability statement.

Here is how to compute the minimum and maximum values for a confidence interval.

Mean	Proportion
CImin = x - SE * CV CImax = x + SE * CV	CImin = p - SE * CV CImax = p + SE * CV

In the table above, x is the sample estimate of the population mean, p is the sample estimate of the population proportion, SE is the standard error, and CV is the critical value (either a z-score or a t-score). And, the confidence interval is an interval estimate that ranges between CI_min and CI_max.

Test Your Understanding

This section presents a sample problem that illustrates how to analyze survey data when the sampling method is proportionate stratified sampling. (In a subsequent lesson, we re-visit this problem and see how stratified sampling compares to other sampling methods.)

Problem 1

At the end of every school year, the state administers a reading test to a sample of third graders. The school system has 20,000 third graders, half boys and half girls.

This year, a proportionate stratified sample was used to select 36 students for testing. Because the population is half boy and half girl, one stratum consisted of 18 boys; the other, 18 girls. Test scores from each sampled student are shown below:

Boys	50, 55, 60, 62, 62, 65, 67, 67, 70, 70, 73, 73, 75, 78, 78, 80, 85, 90
Girls	70, 70, 72, 72, 75, 75, 78, 78, 80, 80, 82, 82, 85, 85, 88, 88, 90, 90

Using sample data, estimate the mean reading achievement level in the population. Find the margin of error and the confidence interval. Assume a 95% confidence level.

Solution: To solve this problem, we follow the seven-step process described above.

• Estimate the population mean. To compute the overall sample mean, we need to compute the sample means for each stratum. The stratum mean for boys is equal to:

  x_boys = Σ ( x_i ) / n

  x_boys = ( 50 + 55 + 60 + ... + 80 + 85 + 90 ) / 18 = 70

  The stratum mean for girls (x_girls) is computed similarly. It is equal to 80. Therefore, overall sample mean is:

  x = Σ( N_h / N ) * x_h

  x = ( 10,000 / 20,000 ) * 70 + ( 10,000 / 20,000 ) * 80

  x = 75

  Therefore, based on data from the sample strata, we estimate that the mean reading achievement level in the population is equal to 75.
• Compute sample variance within strata. We need to compute the sample variance within each stratum, so we can compute the standard error in the next step. For boys, the within-stratum sample variance is equal to:

  s²_boys = Σ ( x_i - x_h )² / ( n_h - 1 )

  s²_boys = [ (50 - 70)² + (55 - 70)² + (60 - 70)² + ... + (85 - 70)² + (90 - 70)² ] / 17 = 105.41

  The within-stratum sample variance for girls (s²_girls) is computed similarly. It is equal to 45.41.
• Compute standard error. The standard error measures the variability of our sample estimate of the population mean. We will use standard error to compute the margin of error and to define a confidence level.

  SE = (1 / N) * sqrt { Σ [ N²_h * ( 1 - n_h/N_h ) * s²_h / n_h ] }

  SE = (1 / 20,000) * sqrt { [ 100,000,000 * ( 1 - 18/10,000 ) * 105.41 / 18 ] + [ 100,000,000 * ( 1 - 18/10,000 ) * 45.41 / 18 ] }

  SE = (1 / 20,000) * sqrt { 99,820,000 * 105.41 / 18 ] + [ 99,820,000 * 45.41 / 18 ] } = 1.45

  Thus, the standard error of the sampling distribution of the mean is 1.45.
• Select a confidence level. In this analysis, the confidence level is defined for us in the problem. We are working with a 95% confidence level.
• Find the critical value. The critical value is a factor used to compute the margin of error. To find the critical value, we take these steps.
  • Compute alpha (α):

    α = 1 - (confidence level / 100)

    α = 1 - 95/100 = 0.05

  • Find the critical probability (p*):

    p* = 1 - α/2 = 1 - 0.05/2 = 0.975

  • Since the sample size (n = 36) is greater than 30, we can express the critical value as a z-score. The critical value is the z-score having a cumulative probability equal to 0.975. From the Normal Distribution Calculator, we find that the critical value is about 1.96.

• Compute the margin of error (ME):

  ME = critical value * standard error

  ME = 1.96 * 1.45 = 2.84

• Specify the confidence interval. The minimum and maximum values of the confidence interval are:

  CI_min = x - SE * CV = 75 - 1.45 * 1.96 = 72.16

  CI_max = x + SE * CV = 75 + 1.45 * 1.96 = 77.84

In summary, here are the results of our analysis. Based on sample data, we estimate that the population mean is 75. Given a 95% confidence level, the margin of error around that estimate is 1.45; and the 95% confidence interval is 72.16 to 77.84.