Sample Proportion, PDF

What Is a Sample Proportion?

The sample proportion (p̂) defines the proportion of individuals in a sample with a specific characteristic or trait. As a statistic and random variable denoted with a capital P or (P̂), the sample proportion varies from sample to sample in one way or another that cannot be predicted with certainty. It has a mean (μPˆ) and a standard deviation (σPˆ).  Being used in statistics and survey research, it may equal the population proportion, or it may not. When researchers and statisticians work with large samples, the sample proportion is almost normally distributed, with mean μ=p and standard deviation σ=pq/n.

For example, if each face of an ordinary dice has an equal chance of landing on top when it is rolled, it is considered fair or balanced. So, the proportion of times a three evident in a huge number of tosses is expected to be approximately ⅙ or 0.16. Suppose the sample proportion of 0.15 came from rolling the dice 2,400 times instead of only 240 times. The probability that a fair dice would create a proportion of 0.15 or less is 0.0139. To interpret this result, a sample proportion of 0.15 or less in only about 1.14% is expected of all samples of size 2,400, indicating a piece of solid evidence that the dice is not fair.

Definitions of Concepts and Terms 

There are numerous concepts and terms related to proportion tests used in statistics and survey research. Be well-informed on the useful definitions of these terms so that you can easily work on your sampling methods and other essential tools to accomplish your research.

Proportion: It shows the percentage of individuals or items that fall into a certain category. For example, the proportion of individuals who follow a specific musical group might be the target. There are two categories shown here, to follow or not to follow the musical group. Population Proportion: Referred to as π (pi), the population proportion is defined as where f is the number of frequencies in the population that fall into the target category and N is the total number of units or individuals in the population. Confidence Interval: It is a range of estimates for an unidentified parameter. It is the mean of your estimate plus and minus the estimate variation. It is used by statisticians to determine the uncertainty in a sample variable.Sample Variance: This term is the average squared deviations from the mean as it reflects variability in a distribution. It measures the average distance or spread from the mean. Standard Deviation or Standard Error: If you need to know how dispersed the data is in relation to the mean, a standard deviation or standard error helps you understand the measurements of the distributed data. So, when there is more distributed data, there is also more standard deviation of the data. To determine standard deviation, search for the mean, find the square of its distance to the mean for each data point, sum up the values from the second step, and divide it by the number of data points. Sampling Variability: This phenomenon is when survey researchers take random samples from a population repeatedly as the sample proportion value varies from sample to sample. If you need to construct a graph with varying sample proportions, the observational units of this graph are the samples, and the variable is the sample proportion. Sampling Distribution of the Proportion: If you need to collect many samples and for each sample compute the sample proportion, you could make a distribution called the sampling distribution of the proportion. For instance, you collect 50 samples of size 100 and measure the proportion for each sample. You would have 50 different sample proportions each based on 100 observations. If you create a frequency distribution of these 50 proportions, it describes the sampling distribution of the proportion. Variance Error: It is the statistical variability of scores made by the influence of variables such as measurement error, sampling error, and error due to variability. To calculate the variance error, add the squares of the differences of each data point from the mean and divides them by the sample size of the number of observations. Binomial Distribution: Also known as the Bernoulli distribution, the binomial distribution is the probability distribution of a binomial random variable. A binomial random variable is the number of successes x in n repeated trials of a binomial experiment or study. It is used to get the probability of observing x successes in N trials, with the success probability on a single trial depicted by p.

Types of Proportions 

Generally, proportions are based on ratios (a:b) and fractions (a/b). These concepts are integral in business, mathematics, and statistics. What are the different types of proportions? 

1. Direct Proportion

It defines the direct relationship between two quantities. Simply, if one quantity increases, the other quantity increases as well. If you are dealing with quantities that are directly related or in direct proportion, use this rule. For instance, 24 x 240 = 12 x 480. 

2. Inverse Proportion

It portrays the indirect relationship between two quantities. So, if one quantity increases, the other one decreases, or when the absolute value or magnitude of one variable gets bigger, the absolute value or magnitude of another variable gets smaller. This type of proportion is written as y ∝ 1/x.

3. Compound Proportion

This type of proportion involves two or more quantities. The quantities could be directly related or inversely related or both. There are some rules to compute compound proportion depending on the cases you are working on. If the first quantity and second quantity are directly related and the second quantity and third quantity are also directly related, then the formula is a x b/ c = d x e /x. If the first quantity and second quantity are directly related and the second quantity and third quantity are inversely related, the formula is b x c/a=e x x/d

4. Continued Proportion

This type of proportion is when an ordered set of numbers like the ratio between any two successive terms is the same or if the ratio between the first and the second is equal to the ratio between the second and the third one. For example, a, b, and c are in continued proportion, if a : b = b : c

How to Calculate Sample Proportion

Use the sampling distribution of the sample proportion to calculate the probability that your sample proportion lies within a specific range: P(p₁ < p̂ < p₂), P(p₁ > p̂), or P(p₁ < p̂). Thus, use the sample size (n), population proportion (p), and the specified proportions range. If you are not aware of the required sample size, use an effective sample size calculator.

Step 1: Understand the Sample Proportion Symbol

Familiarize yourself with the symbol of the sample proportion which is denoted by p̂. Count how many successes or characteristics of interest possessed by an individual are in the sample. Divide it by the total sample size.

Step 2: Draw Charts and Graphs

Construct charts and graphs and insert the fundamental data while calculating your sample proportion. Notice that different samples can provide you with different sample proportions. Plot the frequencies of each sample proportion in a histogram so that you can easily observe the behavior of the sample proportion.

Step 3: Check the Conditions for the Sampling Distribution of Proportions

To help you fully estimate the population proportion for the sampling distribution of proportions, you need to carefully check two conditions: randomization condition and independence or 10% condition. The randomization condition is when you check that your data comes from randomly selected samples. The independence condition is when you make sure that the sampled values are independent one from another. You need to consider sample sizes no larger than 10% of the whole population.

Step 4: Find the Mean and Standard Deviation for Sample Proportions

If you need to find the mean and standard deviations for a sample proportion, use the mean and standard deviation formula by calculating the mean or the simple average of the numbers. For each number, subtract the mean and square the result. Measure the mean of those squared differences and take the square root.


What is sample proportion in hypothesis testing?

The sample proportion in hypothesis testing, also known as the one-sample proportion test, is used to calculate an estimation of a population proportion. It sets a comparison of the proportion to a target or reference value and measures a range of values that is mostly to add the proportion of a population. 

What is the probability of a sample proportion?

The probability of a sample proportion is 50% as long as the sampling distribution of the population proportion is symmetrical.

How to interpret proportions in statistics?

Statisticians and survey researchers use the p-value to identify whether the population proportion is statistically different from the hypothesized proportion.

How does sample proportion affect standard deviation?

The standard deviation decreases when the sample size increase since the sample size appears in the denominator of the square root.

What is the purpose of the sampling proportion?

The sampling proportion is carried out to estimate a population proportion that has a particular characteristic or trait. 

What are the common types of sampling methods?

The common types of sampling methods are cluster sampling, convenience sampling, purposive sampling, stratified sampling, probability sampling, systematic sampling, and many others.

Why are sample proportion and sample mean unbiased?

The sample proportion and sample mean are unbiased because they aim at the parameters. Based on the rule, if the distribution center equals the true population value or the parameter, the distribution is identified as unbiased. 

What is the relationship between confidence interval and confidence level?

The confidence level refers to the percentage of times you anticipate getting near to the same estimate if you conduct your study again or repeat the population sample similarly. The confidence interval contains the upper and lower bounds of the estimate you anticipate t0 search at a given confidence level. 

How to find the sample size for a confidence interval proportion?

To find the sample size for population means, use this formula: n=(z×σE)2. On the other hand, use n=p×(1−p)×(zE)2 to find the sample size for population proportions. 

Whether you are one of the market researchers or survey analysts, a sample proportion is a crucial and valuable method to use, especially if you are conducting a survey or an experiment in a large target group or when everyone is involved in the study. Remember the definitions of the concepts, terms, and other components related to proportion tests such as population proportion, confidence level, and standard deviation, as well as the formulas to calculate the mean and standard deviation for sample proportions. To assist you in your statistics and survey research, check out other templates for survey analysis reports, survey forms, market research business plans, and market research reports