How to Calculate Arithmetic Mean of Ungrouped Data?
Mean of ungrouped data is the arithmetic average of individual observations listed without class intervals. It is calculated by dividing the sum of all observations by their number. This page covers the direct formula, assumed mean method, properties, shortcuts and solved numerical examples for quantitative aptitude.
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What Is the Mean of Ungrouped Data?
If the observations are x₁, x₂, x₃, ..., xₙ, then their arithmetic mean is x̄ = (x₁ + x₂ + x₃ + ... + xₙ) ÷ n. For the data 6, 9, 11 and 14, the mean is (6 + 9 + 11 + 14) ÷ 4 = 40 ÷ 4 = 10.
Mean of Ungrouped Data Video Lessons
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Mean by Direct Method
Learn to calculate the arithmetic mean of ungrouped data using the direct method, including the formula, substitution of values, and step-by-step calculation.
Mean of Ungrouped Data Formula & Tricks
Important Formulas
Here, Σx is the sum of all observations and n is the number of observations.
A is a convenient assumed mean and d is the deviation of each observation from A.
Use this relation when the mean and all observations except one are given.
Quick Tricks
When observations are close to a common number, choose that number as A. Add the deviations to A instead of adding every observation directly.
If k is added to every observation, the mean increases by k. If k is subtracted from every observation, the mean decreases by k.
If every observation is multiplied by k, the new mean is k times the original mean. This also applies when all observations are converted using a fixed multiplication factor.
Mean of Ungrouped Data Concepts
Direct Method
Write each observation once, calculate Σx, count the observations as n, and apply x̄ = Σx ÷ n. All observations must use the same unit before the calculation.
Assumed Mean Method
For each observation x, calculate d = x − A. Then use x̄ = A + Σd ÷ n. Positive deviations belong to observations greater than A, while negative deviations belong to observations less than A.
Effect of Changing Observations
If each value x becomes x + k, the new mean is x̄ + k. If each value is multiplied by k, the new mean is kx̄. If one observation changes by k while the number of observations remains n, the mean changes by k ÷ n.
Finding a Missing Observation
First calculate the required total as mean × number of observations. Then subtract the sum of the available observations. The number of observations includes the missing value.
Combined Mean of Two Data Sets
If two groups have sizes n₁ and n₂ and means x̄₁ and x̄₂, then combined mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂). It is not generally the simple average of the two means unless both groups have equal sizes.
Mean of Ungrouped Data Data Table
Use the structured data below while following the explanation and examples.
Methods for Finding the Mean
Comparison of direct and assumed mean calculations for ungrouped data.
| Method | Calculation | Suitable use |
|---|---|---|
| Direct method | x̄ = Σx ÷ n | Observations are few or easy to add |
| Assumed mean method | x̄ = A + Σd ÷ n | Observations are close to a convenient value |
| Missing-value method | Missing value = required total − known total | Mean and all but one observation are given |
Note: In every method, n represents the total number of observations, including a missing observation if one is being found.
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Mean of Ungrouped Data: Revision Points
Key formulas and rules for quick revision.
- Mean = Sum of observations ÷ Number of observations.
- Direct formula: x̄ = Σx ÷ n.
- Assumed mean formula: x̄ = A + Σd ÷ n, where d = x − A.
- Adding k to every observation adds k to the mean.
- Multiplying every observation by k multiplies the mean by k.
- If one value changes by k among n observations, the mean changes by k ÷ n.
- Missing value = Required mean × Total observations − Sum of known observations.
- For two groups, combined mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).
Mean of Ungrouped Data FAQs
What is the mean of 5, 10, 15 and 20?
Mean = (5 + 10 + 15 + 20) ÷ 4 = 50 ÷ 4 = 12.5.
Which formula is used for the mean of ungrouped data?
The formula is x̄ = Σx ÷ n, where Σx is the sum of observations and n is their total number.
How do you find a missing number when the mean is given?
Use Missing value = (Mean × Total number of observations) − Sum of known observations. For a mean of 12 for four numbers with known sum 31, the missing value is 48 − 31 = 17.
What happens to the mean if 6 is added to every observation?
The mean also increases by 6 because adding a constant to every observation adds the same constant to their average.
Can the mean be a decimal when all observations are integers?
Yes. For example, the mean of 2, 3 and 4 is 9 ÷ 3 = 3, while the mean of 1, 2 and 4 is 7 ÷ 3 = 2⅓.
Why is the simple average of two means not always the combined mean?
The groups may have different sizes. A group-size-weighted formula is required: combined mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).