Sampling Distributions

before you start

This lesson builds on key concepts from an Introduction to Statistics course. Specifically, this tutorial assumes familiarity with the foundational idea of sampling distributions.

1. Introduction to Sampling Distributions

A sampling distribution is a way to understand how a statistic (like an average) from a sample can vary. Imagine you want to know the average amount of money customers spend at a store. Instead of asking every customer, you take several small samples and calculate the average for each sample. The collection of these averages forms a sampling distribution.

Sampling distributions help us understand the variability of our sample statistics. This is important because it allows businesses to make predictions and decisions based on sample data, rather than needing to survey an entire population.

Let’s say a company wants to know the average amount of time customers spend on their website. Instead of tracking every single visitor, they take multiple samples of 100 visitors each and calculate the average time spent for each sample. These averages might be slightly different, but together they form a sampling distribution.

Imagine you take 10 different samples of 100 visitors each and calculate the average time spent on the website for each sample. You might get averages like 5 minutes, 5.2 minutes, 4.8 minutes, etc. If you plot these averages on a graph, you’ll see a distribution of these sample averages. This is your sampling distribution. The histogram below shows a sampling distribution of the average time spent on a website using 10 different samples of 100 visitors each.

Understanding sampling distributions helps businesses make better decisions. For example, if the sampling distribution shows that most sample averages are around 5 minutes, the company can be confident that the true average time spent on the website is close to 5 minutes.

key concept

• Population: The entire group you’re interested in, like all visitors to the website.
• Sample: A smaller group selected from the population, like 100 visitors.
• Sample Statistic: This is a number that describes a sample, like the average time spent on a website.
• Sampling Distribution: The distribution of a statistic (like the average) from many samples.

term to know

Sampling Distribution

Distribution of a given statistic based on a random sample, showing how the statistic varies from sample to sample.

1a. Importance of Sampling Distributions

Sampling distributions are crucial in business data analytics for several reasons:

1. Making Predictions: Sampling distributions help businesses make predictions about a population based on sample data. For example, if a company wants to know the average amount customers spend, they can use a sampling distribution to estimate this average without surveying every customer.
2. Understanding Variability: They show how much a sample statistic (like an average) can vary from sample to sample. This helps businesses understand the reliability of their data. For instance, if the average time spent on a website varies little between samples, the company can be more confident in its estimate.
3. Decision Making: Businesses often make decisions based on sample data. Sampling distributions provide a way to measure the uncertainty of these decisions. For example, a company might use a sampling distribution to decide whether a new marketing strategy is effective, based on sample sales data.
4. Quality Control: In manufacturing, sampling distributions are used to monitor product quality. By taking samples from production lines and analyzing the distribution of defects, companies can identify and address quality issues more efficiently.
5. Cost Efficiency: Surveying an entire population can be expensive and time-consuming. Sampling distributions allow businesses to gather insights and make decisions based on smaller, more manageable samples, saving time and resources.

EXAMPLE

A hospital wants to know the average wait time for patients in the emergency room (ER). Instead of tracking every patient, they take several samples of 100 patients each and calculate the average wait time for each sample. The distribution of these averages forms a sampling distribution.

The key aspects are:

• Population: All patients visiting the ER.
• Sample: Groups of 100 patients each.
• Sample Statistic: Average wait time in the ER.
• Sampling Distribution: The distribution of the average wait times from multiple samples.

The hospital took 50 samples of 100 patients each, calculated the average wait time for each patient, and plotted these 50 averages, as shown in the histogram below.



By analyzing the sampling distribution, the hospital can estimate the true average wait time for all patients. This helps in making decisions about staffing, resource allocation, and process improvements.

The sample averages are between 28 and 32 minutes, which suggests that the average wait time for patients in the ER is likely around 30 minutes, with some variation. This histogram helps the hospital understand the typical wait time and how much it can vary.

1b. Sampling Distribution of the Mean (x̄)

Suppose you want to understand the average spending of customers at a store. Instead of asking every single customer, you take several smaller samples and calculate the average spending for each sample. The collection of these sample averages forms what is known as the sampling distribution of the mean ().

The table below shows the collection of these sample means, which forms the sampling distribution of

Sample	Sample Size	Mean (x̄)
1	100
2	100
3	100
...	...	...
30	100

The histogram below is a plot of all the sample means from the table above. The histogram visualizes the distribution of the sample means. The histogram helps you understand the central tendency (average) and variability of the sample means.

The graphic below illustrates the construction of another sampling distribution of Even though the population mean is 82.5, the image visually demonstrates how the means of samples drawn from that population can vary. This sampling distribution of consists of six samples, each with a sample size of 20.

Just as with other distributions we have studied, the sampling distribution of has its own mean, standard deviation, and shape.

Returning to the example of understanding the average spending of customers at a store, no information is available regarding the original population's spending patterns. However, this is acceptable due to the Central Limit Theorem, a statistical concept that offers insights into the sampling distribution of the sample mean ().

The Central Limit Theorem (CLT) provides you with information about the characteristics of the sampling distribution of The CLT tells us:

• Regardless of the shape of the population distribution (whether skewed, uniform, etc.), the sampling distribution of will be approximately normal (bell-shaped) if the sample size is large enough (at least 30 for most distributions).
• The sampling distribution of has its own mean and standard deviation.
• The sampling distribution of will have a mean equal to the population mean.

• The standard deviation of the sampling distribution of is the population standard deviation divided by the square root of the sample size (this is called the standard error).

EXAMPLE

A retail chain wants to understand its average monthly sales across all its 250 stores to make informed business decisions. However, analyzing the average sales across all its stores is time-consuming and resource-intensive. Instead, the company asks you to use a sample-based approach to estimate the average monthly sales for all its 250 stores.

You create 50 samples of sales from 10 stores and find the average of each sample. That is, you create the sampling distribution of shown in the spreadsheet below. The first five rows of the sampling distribution for sales_CLT.xlsx are shown below.

A description of each column is:

• Sample_Number: Identifies the sample (1 to 50).
• Store_1 to Store_10: The sales figures for each of the 10 stores in that particular sample.
• Sample_Means: The average sales calculated from the 10 stores in that sample. The sample means in column L is the sampling distribution of

For this sampling distribution, the number of samples is 50 and the sample size (n) is 10.



Based on the known properties of the sampling distribution of , you can provide the company with an estimate of the average monthly sales across all 250 stores by using:

For this problem, can be found by taking the mean of the sample means in column L. In Excel, in cell N2, enter:

=AVERAGE(L2:L51)

You find that the mean of the sampling distribution of is $4,948. This gives the retail chain a reliable estimate of the average monthly sales across all stores. The mean of the sample means is a more reliable estimate of the population mean because it aggregates information from multiple samples, reducing the impact of any anomalies in individual samples. This is why the retail chain uses the mean of the sample to estimate the overall average monthly sales for all its stores.

You can also find the standard error by entering the following formula in cell N3:

=STDEV.S(AF2:AF51)/SQRT(50)

The standard error is $26. Since the population standard deviation is not known, the Excel formula that is being used to calculate standard error is:

This helps the retail chain understand the precision of their estimate of the population mean, . That is, this value helps the retail chain understand how much the mean of the sample means ($4,948) is expected to vary from the true population mean. A smaller standard error indicates a more precise estimate of the population mean, μ.

watch

Check out this video on using a sample-based approach to estimate the average monthly sales.

terms to know

Sampling Distribution of x̄

A distribution of all possible sample means of a given size from a population, showing how the sample mean varies from sample to sample.

Central Limit Theorem (CLT)

A property that states the sampling distribution of approaches a normal distribution as the sample size becomes large, regardless of the original population’s distribution.

Standard Error

Measure of the variability of a sample statistic (such as ) from the population, calculated as the standard deviation of the sampling distribution of that statistic.

1c. Estimating Population Mean Using Sampling Distributions: A Practical Approach

The mean of the sampling distribution of is theoretically equal to the population mean, μ. However, in practice, there are several reasons why they might not be exactly the same. For instance:

• Sampling Variability. Each sample you take is different, and the sample mean will vary from sample to sample. This variability is natural and expected. The more samples you take, the closer the mean of the sampling distribution will get to the population mean, but with a limited number of samples, there will always be some difference.
• Sample Size. The size of each sample affects the accuracy. Smaller samples tend to have more variability, which can lead to a greater difference between the sample mean and the population mean. Larger samples tend to produce sample means that are closer to the population mean.
• Random Error. Random errors can occur due to various factors, such as measurement errors or random fluctuations in the data. These errors can cause the sample mean to differ slightly from the population mean.
• Finite Number of Samples. In the previous example, you took 50 samples from 10 stores. While 50 is a good sample size, it is finite. If you were to take an infinite number of samples, the mean of the sampling distribution of would converge to the population mean, μ. With a finite number of samples, there will always be some small difference.

1d. Sampling Distribution of the Proportion (p̂)

Imagine you’re working for a company that wants to know the proportion of employees who prefer working remotely versus in the office. Instead of asking every employee, you decide to survey a smaller group, or sample, of employees. The proportion of employees in your sample who prefer working remotely is called . Now, if you took many different samples from your company, each sample would give you a different . The distribution of all these values from different samples is called the sampling distribution of the proportion (p̂).

The table below shows the collection of these sample proportions, which forms the sampling distribution of

Sample	Sample Size	Sample Proportion (p̂)
1	100
2	100
3	100
...	...	...
30	100

The histogram below is a plot of all the sample proportions from the table above. The histogram visualizes the distribution of the sample proportions. The histogram helps you understand the central tendency (average) and variability of the sample proportions.

The histogram above was created with 30 samples of size 100. If you were to take 100,000 samples of size 100, the histogram would look like the following:

The reason the histogram becomes smoother and looks more like a normal distribution with more samples is due to the law of large numbers and the Central Limit Theorem working together. As you take more samples, the average of the sample proportions will converge to the population proportion. This means that with more samples, the overall distribution of sample means will better represent the true population mean.

The Central Limit Theorem states that the distribution of will approach a normal distribution as the sample size increases. However, this also implies that with a larger number of samples, the variability in the sample means decreases, making the distribution appear more normal.

Below are some key points about the sampling distribution of

• Random Sampling. Each sample should be randomly selected to ensure that every employee has an equal chance of being chosen. This helps make your sample representative of the whole company.
  
  
• Sample Size. The number of employees in each sample affects the sampling distribution. Larger samples tend to give more accurate estimates of the true proportion of employees who prefer working remotely.
  
  
• Shape of the Distribution. The sampling distribution of will approximate a normal distribution as the sample size becomes large.
  
  For the sampling distribution of , a common rule of thumb is that the sample size, n, should be large enough such that both np and are greater than 5, where p is the population proportion. This helps ensure that the normal approximation is valid.
  
  
• Mean of the Distribution. The mean of the sampling distribution of is equal to the true population proportion, p. This means that, on average, will be a good estimate of p. The formula for the mean of the sampling distribution of is:
  
  
  
  In the previous example, if you knew the true population proportion, p, was 60%, the mean of the sampling distribution of would be close to p.
  
  
• Standard Deviation (Standard Error): The standard deviation of the sampling distribution of is called the standard error. It measures how much varies from sample to sample. The formula for the standard error of is:
  
  

terms to know

Sampling Distribution of p̂

A distribution of all possible sample proportions of a given sample size from a population, showing how the sample proportion varies from sample to sample.

Law of Large Numbers

A property that states as the size of a sample increases, the sample mean will become closer to the population mean and the sample proportion will become closer to the population proportion.