SciTech-Mathmatics-Probability+Statistics-Population-Sample: Sample Proportion vs. Sample Mean: The Difference

Sample Proportion vs. Sample Mean: The Difference
BY ZACH BOBBITTPOSTED ON MAY 5, 2021

Two terms that are often used in statistics are Sample Proportion and Sample Mean.
Here's the difference between the two terms:

Sample Proportion: The proportion of observations in a sample with a certain characteristic.
Often denoted \(\large p̂\), It is calculated as follows:
$\large p̂ = x / n $

where:
\(\large x\): The number of observations in the sample with a certain characteristic.
\(\large n\): The total number of observations in the sample.
\(\large Sample\ mean\): The average value in a sample.

Often denoted \(\large x\), it is calculated as follows:
$\large x = \sum{x_i}/n $

where:
$\large \sum \(: A symbol that means "sum" \)\large x_i $: The value of the \(\large i\)th observation in the sample
$\large n $: The sample size

Sample Proportion vs. Sample Mean: When to Use Each

The sample proportion and sample mean are used for different reasons:

• Sample proportion :
  Used to understand the proportion of observations in a sample that have a certain characteristic.
  For example, we could use the sample proportion in each of the following scenarios:

  • Politics : Researchers might survey 500 individuals in a certain city,
    to understand what proportion of residents support a certain candidate in an upcoming election.
  • Biology : Biologists may collect data on 100 sea turtles,
    to understand what proportion of them have experienced damage from pollution.
  • Sports : A journalist may survey 1,000 college basketball players,
    to understand what proportion of them shoot left-handed.

• Sample mean :
  Used to understand the average value in a sample.
  For example, we could use the sample mean in each of the following scenarios:

  • Demographics : Economists may collect data on 5,000 households in a certain city to estimate the average annual household income.
  • Botany : A botanist may take measurements on 50 plants from the same species to estimate the average height of the plant in inches.
  • Nutrition : A nutritionist may survey 100 people at a hospital to estimate the average number of calories that residents eat per day.

Depending on the question of interest, it might make more sense to use the sample proportion or the sample mean to answer the question.

Using the Sample Proportion & Sample Mean to Estimate Population Parameters

Both the sample proportion and the sample mean are used to estimate population parameters.

Sample Proportion as an Estimate

We use the sample proportion to estimate a population proportion.
For example, we might be interested in understanding,
what proportion of residents in a certain city support a new law.

• Since it would be too costly and time-consuming to survey all 40,000,000 residents in the city, we instead survey 500 and calculate the proportion of residents in the sample who support the new law.
  • We then use this sample proportion as our best estimate of the proportion of residents in the entire city who suppose the new law.
  • However, since $\large \bm{ it's \ unlikely } that our sample proportion \(\large \bm{ exactly\ matches }\) the population proportion, we often use a \(\large \bm{ confidence\ interval }\) for a proportion – a range of values that we believe $\large \bm{ contains } $ the true population proportion with a certain level of confidence.

Sample Mean as an Estimate

We use the sample mean to estimate a population mean.
For example, we might be interested in understanding,
the average height of a certain species of plants.

• Since it would be too costly and time-consuming to measure the height of all 10,000 plants in a certain region, we instead measure the height of 150 plants and use the sample mean as our best estimate of the population mean.
• However, since $\large \bm{ it's \ unlikely } that our sample mean \(\large \bm{ exactly\ matches }\) the population mean, we often use a \(\large \bm{ confidence\ interval }\) for a mean– a range of values that we believe $\large \bm{ contains } $ the true population mean with a certain level of confidence.

Additional Resources
Confidence Interval for Proportion Calculator
Confidence Interval for Mean Calculator