# Difference Between Sample Mean and Population Mean

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## Introduction

Statistics is a branch of applied mathematics concerned with the collection, description, analysis, and inference of conclusions from numerical data. Differential and integral calculus, linear algebra, and probability theory are all heavily used in the mathematical theories of statistics. Statisticians are particularly interested in learning how to draw valid conclusions about large groups and general events from small sample behaviour and other observable characteristics. These small samples are representative of a subset of a larger group or a small number of occurrences of a common occurrence.

The two major disciplines of statistics are descriptive statistics, which explain the characteristics of sample and population data, and inferential statistics, which uses those properties to test hypotheses and draw conclusions.

Descriptive statistics are a collection of short descriptive coefficients that summarise information.

Descriptive statistics are a series of short descriptive coefficients that summarise a data set, which might be a representation of the complete population or a sample of the population. Measures of central tendency and measures of variability are two types of descriptive statistics (spread). The mean, median, and mode are examples of central tendency measures, while standard deviation, variance, minimum and maximum variables, kurtosis, and skewness are examples of variability measures.

People utilize descriptive statistics to break down difficult-to-understand quantitative findings from a large data set into digestible chunks. The grade point average (GPA) of a student, for example, is an excellent indicator of descriptive statistics. The concept behind a GPA is that it averages data from a variety of exams, classes, and grades to provide a basic understanding of a student's overall academic performance. The average academic success of a student is reflected in their GPA.

The arithmetic mean is one of the best metrics of central tendency in statistics. The arithmetic mean can be computed for a group of data by adding all of them together and dividing the result by the number of observations. In statistics and probability, there are two sorts of means: sample mean and population mean. When the population means are unknown, the sample mean is used to approximate it because they have the same expected value.

The Sample Mean denotes the average of a sample drawn at random from the entire population. The population means is just the average of everyone in the group. To learn more about the distinctions between the sample mean and population mean, see this article.

## Sample Mean vs. Population Mean

In math class, we learned about statistics and did mean median, and mode calculations. These are statistical phrases in the field of mathematics, and I'm sure the subject did not appeal to everyone.

The major distinction between sample and population mean is that sample mean refers to the sample values that have been amassed or collected, whereas population means refers to the population's mean. Though both sample mean and population mean are calculated in the same way, they are denoted by different signs. The sample mean is represented by the symbol or letter x with a bar at the top, whereas the population mean is represented by the Greek word mu.

The average of all the values in a sample is called the "mean." It's computed by adding all of the values together and then dividing the total by the number of values in the sample.

The mean is referred to as the population mean when the presented list represents a statistical population. The letter is generally used to represent it.

When a list represents a statistical sample, the mean is referred to as the sample mean. "X" stands for the sample mean. It's a good approximation of the population mean. A population mean for a sample can be calculated asΣ = x / n, where;

n is the number of observations taken for the study; represents the sum of all the number of observations in the population.

When the data also includes frequency, the mean can be calculated asΣ = f x / n, where;

f denotes the class frequency; x denotes the class value; n denotes the population size, and represents the sum of the products "f" and "x" across all classes.

Similarly, the sample mean is X =Σx / for µ=Σf x / n, where "n" is the number of observations.

It can also be written as X = x1 + x2 + x3 +................xn / n or X = 1/n(x1 + x2 + x3 +................xn)=Σ x / n.

The Sample Mean denotes the average of a sample drawn at random from the entire population. The population means is just the average of everyone in the group. To learn more about the distinctions between the sample mean and population mean, see this article.

## Difference Between Sample Mean and Population Mean in Tabular Form

 Specifications Sample Mean Population Mean Meaning The arithmetic mean of random sample values chosen from the population is known as the sample mean. The population mean is the true average of the entire population. Symbol x̄ (pronounced as x bar) μ (Greek mu) Calculations Easy Difficult Accuracy Low High Standard deviation When the sample mean is used, the result is denoted by (s). When the population mean is used, the result is given by (σ).

## What is Sample Mean?

A sample is a small portion of a larger total. For example, if you want to find out how much people spend on food each year, you probably won't poll more than 300 million people. Instead, you select a subset of that 300 million (maybe a thousand people), which is referred to as a sample. "Average" is another synonym for "mean." In this case, the sample mean would be the average annual cost of food for those thousand persons.

The sample mean is useful because it allows you to approximate the behavior of the entire population without having to survey everyone. Let's say the average annual income in your food sample was \$2400.

The formula for calculating the sample mean is as follows:

X̄ = ( Σ xi ) / n

If that seems difficult, don't worry; it's easier than you think. Remember how to calculate the "average" in elementary math? It's the same thing; the only difference is the notation (i.e. the symbols). Let's divide it down into its constituent parts:

X̄ just stands for the “sample mean”

Σ is summation notation, which means “add up”

Xi “all of the x-values”

N means “the number of items in the sample”

Now all you have to do is enter in the numbers you've been provided and solve them using arithmetic (no algebra is required—you can just plug this into any calculator).

You can come across the following sample mean formula:

X̄ = 1/ n * ( Σ xi )

The setup is slightly different, but algebraically it’s the same formula (if you simplify the formula 1/n * X, you get 1/X).

Finding the sample mean is similar to calculating the average of a collection of integers. You'll see slightly different notation in statistics than you're usually used to, but the math is the same.

All the formula says is, to sum up, all of the numbers in your data set(Σindicates "all of the numbers in the data set"). This post will show you how to manually calculate the sample mean (this is also one of the AP Statistics formulas). However, if you're looking for the sample mean, you'll presumably be looking for other descriptive statistics as well, such as the sample variance or the interquartile range, so you might want to look for it in Excel or another technology. Why? Although the mean calculation is straightforward, if you use Excel, you will only have to enter the information once. After that, you can utilize the numbers to find any statistic you want: not just the one you're looking for.

The variance of the mean sampling distribution. If you're not familiar with the central limit theorem, go back and read The Mean of the Sampling Distribution of the Mean.

The sample mean sampling distribution is a probability distribution of all sample means. Assume you had 1,000 people and sampled 5 people at a time to get their average height. If you maintained taking samples (i.e., you repeated the sampling a thousand times), the mean of all of your samples would eventually become the mean of all of your samples.

1. Equal the population mean, μ
2. Looks like a normal distribution curve.

The variance of this probability distribution indicates how far your data is dispersed around the mean. The sample mean will be more closely related to the population mean as the sample size grows. In other words, as N increases, the variance decreases. The variance should be zero when the sample mean matches the population mean.

The variance of the sample distribution of the mean can be calculated using the following formula:

σ2M = σ2 / N,

where:

σ2M = variance of the sampling distribution of the sample mean.

σ2 = population variance.

## What is Population Mean?

The population mean is a group characteristic's average. "All people residing in the United States" or "all dog owners in Georgia" are examples of groups that could be made up of people, items, or things. A distinguishing feature is only a point of interest. Consider the following scenario:

The average GPA in a class of 1,013 students is 3.1.

The average weight of dogs seen at a particular veterinarian clinic is 38 pounds.

On average, books in a school’s public library are checked out seven times per year.

It is quite rare in statistics to be able to calculate the population mean. That's because polling a full population is usually either too expensive or too time-consuming. One veterinarian's office, for example, may retain weight data for all of the pets that come through the door. This allows you to calculate the average weight of a dog for that practice (i.e. the population mean for that practice). However, if you worked for a pet food company and wanted to know the average weight of a dog, you wouldn't be able to track down and weigh all of the US's 70 to 80 million dogs. You'd have to weigh a sample of dogs (a small part of the whole population).

The population mean symbol is μ.

The formula to find the population mean is:

μ = (Σ * X)/ N

where:

Σ means “the sum of.”

X = all the individual items in the group.

N = the number of items in the group.

## The Main Difference between Sample Mean and Population Mean in Points

The following points explain the significant disparities between the sample mean and the population mean in greater detail:

• The sample mean is the arithmetic mean of random sample values collected from the population. The population mean is the arithmetic mean of the total population.
• The sample is denoted by the letter x̄. (pronounced as an x bar). The population mean, on the other hand, is abbreviated as μ (Greek term mu).
• While calculating the sample mean is simple due to the limited number of factors provided, it takes very little time. In contrast to the population mean, which is difficult to calculate since many elements in the population require a long time to calculate.
• A population mean has a better level of accuracy than a sample mean. By increasing the number of observations, the accuracy of a sample mean can be improved.
• In population mean, 'N' represents elements of the population. The 'n' in the sample mean, on the other hand, denotes the sample size.
• The standard deviation is represented by the letters' when it is calculated using the sample mean. The population mean, on the other hand, is represented by sigmaσ when employed in the computation of standard deviation.

## Conclusion

Although both means are calculated using the same procedure (sum of all observations divided by the number of observations), there is a significant difference in how they are represented. A sample mean is denoted by the letters x or M, whereas the population mean is denoted by the letters. The population mean is an unknown constant, whereas the sample mean is a random variable.