How to Estimate a Population Total from a Cluster Sample

This lesson describes how to estimate a population total, given survey data from a cluster sample. A good analysis should provide two outputs:

• A point estimate of the population total.
• 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

A good analysis of survey data from a cluster sample includes eight steps:

• Estimate a population parameter (in this case, the population total).
• Compute sample variance within each cluster.
• Compute sample variance between each cluster.
• 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 cluster sampling.

Estimating a Population Total

The first step in the analysis is to develop a point estimate for the population total. Before we can accomplish this objective, we need to compute a mean score or a proportion for each sampled cluster.

Use this formula to compute the sample means:

Sample mean in cluster h = x_h = Σx_h / m_h

where Σx_h is the sum of all the sample observations in cluster h, and m_h is the number of sample observations in cluster h.

Once we know the sample mean in each cluster, we can estimate the population total (t) from the following formula:

Population total = t = N/n * ΣM_h * x_h

where N is the number of clusters in the population, n is the number of clusters in the sample, M_h is the number of observations in the population from cluster h, and x_h is the sample mean from cluster h.

A proportion is a special case of the mean. It represents the number of observations that have a particular attribute divided by the total number of observations in the group. Use this formula to compute the proportion for each sampled cluster:

p_h = m'_h / m_h

where p_h is a sample estimate of the population proportion for cluster h, m'_h is the number of sample observations from cluster h that have the attribute, and m_h is the total number of sample observations from cluster h.

Once we have estimated a sample proportion for each cluster, we can estimate a population total:

Population total = t = N/n * ΣM_h * p_h

where t is an estimate of the number of elements in the population that have a specified attribute, N is the number of clusters in the population, n is the number of clusters in the sample, M_h is the number of observations from cluster h in the population, and p_h is the sample proportion from cluster h.

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 Clusters

If you are using one-stage cluster sampling, you can skip this step. But if you are using two-stage cluster sampling, you will need to compute the variance within each sampled cluster.

For a mean score, the variance within each cluster can be estimated from sample data as:

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

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

For a proportion, the variance within each cluster can be estimated as:

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

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

Computing Variance Between Clusters

Use the following formula to estimate the variance of total scores between sampled clusters (s²_b):

s²_b = Σ ( t_h - t/N )² / ( n - 1 )

where s²_b is a sample estimate of the variance between sampled clusters, t_h is the total from cluster h, t is the sample estimate of the population total, N is the number of clusters in the population, and n is the number of clusters in the sample.

Note: If you are working with proportions, t_h is:

t_h = M_h * p_h

where M_h is the number of population elements in cluster h, and p_h is the observed proportion in cluster h.

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 population total from a cluster sample, the standard error (SE) of the estimate is:

SE =	N * sqrt { [ ( 1 - n/N ) / n ] * s2b/n +
	N/n * Σ ( 1 - mh/Mh ) * M2h * s2h/mh ) }

where N is the number of clusters in the population, n is the number of clusters in the sample, s²_b is a sample estimate of the variance between clusters, m_h is the number of elements from cluster h in the sample, M_h is the number of elements from cluster h in the population, and s²_h is a sample estimate of the population variance in cluster h.

With one-stage cluster sampling, the formula for the standard error reduces to:

SE = N * sqrt { [ ( 1 - n/N ) / n ] * s²_b/n }

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 cluster sample, use this equation:

    df = Σ ( m_h - 1 )

    where m_h is the number of sample observations from cluster 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.

Sample Problem

This section presents a sample problem that illustrates how to estimate a population total when the sampling method is one-stage cluster sampling.

Example 1

A botanist divides a field into 1000 equal-size plots. In each plot, he plants 100 clover seeds; and ultimately, each seed sprouts. The botanist randomly selects 20 plots and counts the number of four-leaf clovers in each sampled plot. His findings appear below:

0, 1, 1, 2, 2, 2, 3, 3, 3, 3 3, 3, 3, 3, 4, 4, 4, 4, 4, 8

Using sample data, estimate the total number of four-leaf clovers in the field. 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 total. Before we can estimate the population total, we need to first estimate the sample mean for each cluster. The formula for a cluster mean is:

  Sample mean in cluster h = x_h = Σx_h / m_h

  where Σx_h is the sum of all the sample observations in cluster h, and m_h is the number of sample observations in cluster h.

  Using the above formula, we can compute a sample mean for each of the 20 sampled plots:

  Mean_{plot 1} = x₁ = Σx₁ / n₁ = 0/100 = 0

  Mean_{plot 2} = x₂ = Σx₂ / n₂ = 1/100 = 0.01

  . . .

  Mean_{plot 19} = x₁₉ = Σx₁₉ / n₁₉ = 4/100 = 0.04

  Mean_{plot 20} = x₂₀ = Σx₂₀ / n₂₀ = 8/100 = 0.08

  Given the sample means within strata, we can estimate the population total (t) from the following formula:

  t = N/n * ΣM_h * x_h

  t = 1000/20 * ( 100 * 0 + 100 * 0.01 + ... + 100 * 0.04 + 100 * 0.08 )

  t = 50 * 60 = 3000

  Therefore, based on sampled data, we estimate that there are 3000 four-leaf clovers in the field.
• Compute sample variance within each cluster. If our problem involved two-stage cluster sampling, we would need to compute sample variance within each cluster. But since our problem uses one-stage cluster sampling, we don't need to compute variance within clusters.
• Compute sample variance between clusters. We use the following formula to estimate the variance of total scores between sampled clusters (s²_b):

  s²_b = Σ ( t_h - t/N )² / ( n - 1 )

  s²_b = 1/19 * Σ ( t_h - 3000/1000 )² = 0.05263 * Σ ( t_h - 3 )²

  s²_b = 0.05263 * [(0-3)² + (1-3)² + ... + (4-3)² + (8-3)²]

  s²_b = 0.05263 * [9 + 4 + 4 + 1 + 1 + 1 + . . . + 1 + 1 + 1 + 1 + 1 + 25]

  s²_b = 0.05263 * 50 = 2.63

• Compute standard error. With one-stage cluster sampling, the standard error (SE) of the estimate is:

  SE = N * sqrt [ ( 1 - n/N ) * s²_b/n ]

  SE = 1000 * sqrt [ ( 1 - 20/1000 ) * 2.63/20 ]

  SE = 1000 * sqrt ( 0.098 * 0.1315 )

  SE = 1000 * sqrt (0.12887) = 1000 * 0.359 = 359

  Thus, the standard error of the sampling distribution of the total is 359.

• 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.
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• 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 = 20) is less than 30, we will express the critical value as a t-score with degrees of freedom (df) equal to:

    df = n - 1 = 20 - 1 = 19

    Thus, the critical value is the t-score with 19 degrees of freedom that has a cumulative probability equal to 0.975. From the t-Distribution Calculator, we find that the critical value is about 2.09.

• Compute the margin of error (ME):

  ME = critical value * standard error

  ME = 2.09 * 359 = 750

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

  CI_min = x - SE * CV = 3000 - 750 = 2250

  CI_max = x + SE * CV = 3000 + 750 = 3750

In summary, here are the results of our analysis. Based on sample data, we estimate that there are 3000 four-leaf clovers in the field. Given a 95% confidence level, the margin of error around that estimate is 750; and the 95% confidence interval is 2250 to 3750.