Sample mean represents the measure of the center of the data. Any population’s mean is estimated using the sample mean. In many of the situations and cases, we are required to estimate what the whole population is doing, or what all are the factors going throughout the population, without surveying everyone in the population. In such cases sample mean is useful. An average value found in a sample is termed the sample mean. the sample mean so calculated is used to find the variance and thereby the standard deviation. Let us see the sample mean formula and its applications in the upcoming sections.
What Is Sample Mean Formula?
The sample mean formula is used to calculate the average value of the given sample data. It is necessary sometimes to calculate the average of the sample terms rather than expressing them with actual terms. The sample mean formula can be expressed as the ratio of the sum of terms to the number of terms. i.e.,
Sample Mean Formula
The sample mean formula is written as
Where,
- ∑xi
- = sum of terms
- n = number of terms
Let us see the applications of the sample mean formula in the section below.
Examples on Sample Mean Formula
Example 1: Find the sample mean of 60, 57, 109, 50.
Solution:
To find: Sample mean
Sum of terms = 60 + 57 + 109 + 50 = 276
Number of terms = 4
Using sample mean formula,
mean = (sum of terms)/(number of terms)
mean = 276/4 = 69
Answer: The sample mean of 60, 57, 109, 50 is 69.
Example 2: Five friends having heights of 110 units, 115 units, 109 units, 112 units, and 114 units respectively. Find their sample mean height.
Solution:
To find: Sample mean height
Sum of all heights = 110 + 115 + 109 + 112 + 114 = 560
Number of person = 5
Using sample mean formula,
mean = (sum of terms)/(number of terms)
mean =560/5
= 112 units
Answer: The sample mean height of five friends is 112 units.
Example 3: Five children having different time slots of finishing their homework 30 minutes, 60 minutes, 45 minutes, 40 minutes, and 90 minutes respectively. Find their sample mean time.
Solution:
To find: Sample mean time
Sum of all time slots= 30 + 40 + 45 + 60 + 90 = 265
Number of children = 5
Using sample mean formula,
mean = (sum of terms)/(number of terms)
mean =265/5
= 53 minutes
Answer: The sample mean time of five children is 53 minutes.
FAQs on Sample Mean Formula
How to Calculate the Sample Mean Using Sample Mean Formula?
The general formula for calculating the sample mean is given by x̄ = ( Σ xi ) / n. Here, x̄ represents the sample mean, xi refers all X sample values and n stands for the number of sample terms in the data set. When calculating the sample mean the following steps can be considered:
- Add the total sample items
- Divide the total of sample items by the number of samples.
- The obtained result is the sample mean.
What Does n Represent in the Sample Mean Formula?
The general sample mean formula for calculating the sample mean is expressed as x̄ = ( Σ xi ) ÷ n. Here, x̄ denotes the average value of the samples or sample mean, xi refers all X sample values and ‘n’ stands for the number of sample terms in the given data.
How To Add Up the Sample Items Using the Sample Mean Formula?
To add up the sample items or to get the sum up value of all the sample items we follow the following pattern:
∑xi=(x1+x2+x3+⋯+xn).
Sample Mean: Symbol (X Bar), Definition, Standard Error
Sample Mean Symbol and Definition
The sample mean symbol is x̄, pronounced “x bar”.
The sample mean is an average value found in a sample.
A sample is just a small part of a whole. For example, if you work for polling company and want to know how much people pay for food a year, you aren’t going to want to poll over 300 million people. Instead, you take a fraction of that 300 million (perhaps a thousand people); that fraction is called a sample. The mean is another word for “average.” So in this example, the sample mean would be the average amount those thousand people pay for food a year.
The sample mean is useful because it allows you to estimate what the whole population is doing, without surveying everyone. Let’s say your sample mean for the food example was $2400 per year. The odds are, you would get a very similar figure if you surveyed all 300 million people. So the sample mean is a way of saving a lot of time and money.
Formula
The sample mean formula is:
x̄ = ( Σ xi ) / n
If that looks complicated, it’s simpler than you think (although check out our tutoring page if you need help!). Remember the formula to find an “average” in basic math? It’s the exact same thing, only the notation (i.e. the symbols) are just different. Let’s break it down into 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 it’s just a matter of plugging in numbers that you’re given and solving using arithmetic (there’s no algebra required—you can basically plug this in to any calculator).
You might see the following alternate sample mean formula:
x̄ = 1/ n * ( Σ xi )
The set up is slightly different, but algebraically it’s the same formula (if you simplify the formula 1/n * X, you get 1/X).
How to Find the Sample Mean
Finding the sample mean is no different from finding the average of a set of numbers. In statistics you’ll come across slightly different notation than you’re probably used to, but the math is exactly the same.
All that formula is saying is add up all of the numbers in your data set ( Σ means “add up” and xi means “all the numbers in the data set). This article tells you how to find the sample mean by hand (this is also one of the AP Statistics formulas). However, if you’re finding the sample mean, you’re probably going to be finding other descriptive statistics, like the sample variance or the interquartile range so you may want to consider finding the sample mean in Excel or other technology. Why? Although the calculation for the mean is fairly simple, if you use Excel then you only have to enter the numbers once. After that, you can use the numbers to find any statistic: not just the sample mean.
How to Find the Sample Mean: Steps
Sample Question: Find the sample mean for the following set of numbers: 12, 13, 14, 16, 17, 40, 43, 55, 56, 67, 78, 78, 79, 80, 81, 90, 99, 101, 102, 304, 306, 400, 401, 403, 404, 405.
Step 1:Add up all of the numbers:
12 + 13 + 14 + 16 + 17 + 40 + 43 + 55 + 56 + 67 + 78 + 78 + 79 + 80 + 81 + 90 + 99 + 101 + 102 + 304 + 306 + 400 + 401 + 403 + 404 + 405 = 3744.
Step 2:Count the numbers of items in your data set. In this particular data set there are 26 items.
Step 3:Divide the number you found in Step 1 by the number you found in Step 2. 3744/26 = 144.
That’s it!
Tip: If you have to show working out on a test, just place the two numbers into the formula. Step 1 gives you the σ and Step 2 gives you n:
x = ( Σ xi ) / n
= 3744/26
= 144
Variance of the sampling distribution of the sample mean
variance of the sampling distribution of the mean. If you aren’t familiar with the central limit theorem, you may want to read the previous article: The Mean of the Sampling Distribution of the Mean.
The sampling distribution of the sample mean is a probability distribution of all the sample means. Let’s say you had 1,000 people, and you sampled 5 people at a time and calculated their average height. If you kept on taking samples (i.e. you repeated the sampling a thousand times), eventually the mean of all of your sample means will:
- Equal the population mean, μ
- Look like a normal distribution curve.
The variance of this probability distribution gives you an idea of how spread out your data is around the mean. The larger the sample size, the more closely the sample mean will represent the population mean. In other words, as N grows larger, the variance becomes smaller. Ideally, when the sample mean matches the population mean, the variance will equal zero.
The formula to find the variance of the sampling distribution of the mean is:
σ2M = σ2 / N,
where:
σ2M = variance of the sampling distribution of the sample mean.
σ2 = population variance.
N = your sample size.
Sample question: If a random sample of size 19 is drawn from a population distribution with standard deviation α = 20 then what will be the variance of the sampling distribution of the sample mean?
Step 1: Figure out the population variance. Variance is the standard deviation squared, so:
σ2 = 202 = 400.
Step 2: Divide the variance by the number of items in the sample. This sample has 19 items, so:
400 / 19 = 21.05.
That’s it!
How to Calculate Standard Error for the Sample Mean: Overview
The standard error of the mean of a sample is equal to the standard deviation for the sample. The difference between standard error and standard deviation is that with standard deviations you use population data (i.e. parameters) and with standard errors you use data from your sample. You can calculate standard error for the sample mean using the formula:
SE = s / √(n)
SE = standard error, s = the standard deviation for your sample and n is the number of items in your sample.
Calculate Standard Error for the Sample Mean: Steps
Example: Find the standard error for the following heights (in cm): Jim (170.5), John (161), Jack (160), Freda (170), Tai (150.5).
Step 1: Find the mean (the average) of the data set: (170.5 + 161 + 160 + 170 + 150.5) / 5 = 162.4.
Step 2: Calculate the deviation from the mean by subtracting each value from the mean you found in Step 1.
170.5 – 162.4 = -8.1
161 – 162.4 = 1.4
160 – 162.4 = 2.4
170 – 162.4 = -7.6
150.5 – 162.4 = 11.9
Step 3: Square the numbers you calculated in Step 2:
-8.1 * -8.1 = 65.61
1.4 * 1.4 = 1.96
2.4 * 2.4 = 5.76
-7.6 * -7.6 = 57.76
11.9 * 11.9 = 141.61
Step 4: Add the values you calculated in Step 3:
65.61 + 1.96 + 5.76 + 57.76 + 141.61 = 272.7
Step 5: Divide the number you found in Step 4 by your sample size – 1. There are five items in the sample, so n-1 = 4:
272.7 / 4 = 68.175.
Step 6: Take the square root of the number you found in Step 5. This is your standard deviation.
√(68.175) = 8.257
Step 6: Divide the number you calculated in Step 6 by the square root of the sample size (in this sample problem, the sample size is 5):
8.257 / √(5) = 8.257 / 2.236 = 3.693
That’s how to calculate the standard error for the sample mean!
Tip: If you’re asked to find the “standard error” for a sample, in most cases you’re finding the sample error for the mean using the formula SE = s/√n. There are different types of standard error though (i.e. for proportions), so you may want to make sure you’re calculating the right statistic.
Usage
The sample mean formula calculates the average value of sample data. For example, to calculate the sample mean of the sample data [1,2,5,6,7,9], sum together all of the data points and divide by the number of elements.
The mean of the dataset [1,1,2,3,5] is 2.4.
Sample Problems
Problem 1. Find the sample mean of the data 15, 20, 72, 43, and 21.
Solution:
We have the sample, 15, 20, 72, 43, 21
Sum of terms(S) = 15 + 20+ 72 + 43 + 21 = 171
Number of terms(n) = 5
Using the formula for sample mean, we get
x̄ = S/n
= 171/5
= 34.2
Problem 2. Find the sample mean of the data 13, 31, 27, 72, 16, and 67.
Solution:
We have the sample, 13, 31, 27, 72, 16, 67
Sum of terms(S) = 13 + 31+ 27 + 72 + 16 + 67 = 171
Number of terms(n) = 6
Using the formula for sample mean, we get
x̄ = S/n
= 226/6
= 37.66
Problem 3. Find the sample mean of the data 42, 53, 92, 31, 56, 110, and 63.
Solution:
We have the sample, 42, 53, 92, 31, 56, 110, 63
Sum of terms(S) = 42 + 53+ 92 + 31 + 56 + 110 + 63 = 447
Number of terms(n) = 7
Using the formula for sample mean, we get
x̄ = S/n
= 447/7
= 63.85
Problem 4. Find the number of terms in the sample if their sum and mean are 132 and 22 respectively.
Solution:
We have, S = 132, x̄ = 22
Using the formula for sample mean, we get
x̄ = S/n
=> 22 = 132/n
=> n = 132/22
=> n = 6
Problem 5. Find the number of terms in the sample if their sum and mean are 315 and 35 respectively.
Solution:
We have, S = 315, x̄ = 35
Using the formula for sample mean, we get
x̄ = S/n
=> 35 = 315/n
=> n = 315/35
=> n = 9
Problem 6. In a class of 10 students, the data for marks of all the students is 20, 25, 30, 35, 40, 45, 50, 55, 60, 65. Find the average marks for the class.
Solution:
We have the sample, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65
Sum of terms(S) = 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55 + 60 + 65 = 425
Number of terms(n) = 10
Using the formula for sample mean, we get
x̄ = S/n
= 425/10
= 42.5
Problem 7. Seven workers worked at a construction site for 15, 30, 45, 60, 75, 90, and 105 days respectively. Find the average number of days they worked.
Solution:
We have the sample, 15, 30, 45, 60, 75, 90, 105
Sum of terms(S) = 15 + 30 + 45 + 60 + 75 + 90 + 105 = 420
Number of terms(n) = 7
Using the formula for sample mean, we get
x̄ = S/n
= 420/7
= 60
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