A sample is a subset of a bigger population of data. The sample mean formula is used to get the average value of a set of sample data. It is defined as the average of the sample's measurements. If the sample is chosen at random, the sample mean may be used to calculate the population mean. It represents the assessment of the data center and is used to determine the mean of any population.
Sample Mean Definition
The sample mean is a measure of central tendency that represents the average value of a sample. It's calculated by summing all the observed values and then dividing by the number of observations in the sample. The sample mean formula is shown in the image added below:
Sample MeanSample mean for a data set is defined as the sum of all the terms divided by the total number of terms. It is denoted by the symbol x̄ and is calculated using the formula:
x̄ = Σxi / n
where,
- Σxi is Sum of Terms in the Sample
- n is Number of Terms in Sample
Key Points
- The sample mean provides a single value that summarizes the central tendency of the data.
- It is sensitive to outliers, as extreme values can significantly affect the mean.
- The sample mean is commonly used in various statistical analyses and is an essential concept in descriptive statistics.
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
1. Given the sample data: 12, 15, 20, 22, 30. Calculate the sample mean.
2. For the sample data: 5, 7, 7, 8, 10, 10, 10. Find the sample mean.
3. For a set of grouped data with the following midpoints and frequencies:
- Midpoints: 10, 20, 30
- Frequencies: 4, 5, 6
Compute the sample mean.
4. Consider the following sample data: -5, -3, 2, 4, 1. What is the sample mean?
5. The following data represents the number of units sold by a company over 10 days: 15, 22, 19, 30, 25, 18, 27, 20, 23, 17. Calculate the sample mean.
6. For the sample data: 3.5, 4.2, 5.8, 2.9, 4.6. Calculate the sample mean.
7. A sample of 6 temperatures recorded in Celsius is: 22.3, 23.1, 21.8, 22.7, 23.5, 24.2. Calculate the sample mean and discuss how it reflects the central tendency of the data.
8. For the temperature readings recorded over a week: 18.2°C, 20.5°C, 21.1°C, 19.8°C, 20.3°C, 21.0°C, 18.9°C. What is the sample mean temperature?
9. A person tracks their daily steps for a week: 8500, 9200, 7800, 8700, 9100, 9500, 8900. Calculate the sample mean number of steps.
10. For the following monthly incomes (in dollars): 2200, 2500, 2700, 2400, 2600. Find the sample mean income.
Conclusion
A simple method of summarizing data is to get the average value from a selection of observations; this is where sample mean comes in handy. It aids several statistical investigations, giving light on central tendencies. The application problems illustrate use across diverse datasets and contexts. Though a nice statistic, it should be noted that sample mean has limitations, such as being too susceptible to extreme values. For effective data analysis, a sound understanding of calculations behind the sample mean and its various interpretations is important.
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