The mean of a distribution is 10 and the standard deviation is 5. What is the value of variance coefficient? A. 50% B. 100% C. 150% D. 200%
A
Understanding the Coefficient of Variance
The question asks for the value of the "variance coefficient," which is more commonly known as the Coefficient of Variance (CV). The Coefficient of Variance is a statistical measure of the relative dispersion of data points around the mean. It is a standardized measure that allows comparison of variability between datasets with different means.
Coefficient of Variance Formula
The formula to calculate the Coefficient of Variance is:
\(\text{CV} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100\%\)
Sometimes it is also expressed as a decimal, but the question options are in percentages, so we will use the percentage form.
Calculating the Coefficient of Variance
We are given the following values:
- Mean (\(\bar{x}\)) = 10
- Standard Deviation (\(\sigma\)) = 5
Now, we can plug these values into the formula:
\(\text{CV} = \left( \frac{5}{10} \right) \times 100\%\)
First, calculate the ratio of the standard deviation to the mean:
\(\frac{5}{10} = 0.5\)
Next, multiply this ratio by 100% to express it as a percentage:
\(\text{CV} = 0.5 \times 100\%\)
\(\text{CV} = 50\%\)
Interpreting the Result
The Coefficient of Variance for this distribution is 50%. This means the standard deviation is 50% of the mean. A higher CV indicates greater relative variability.
Matching with Options
Let's compare our calculated value with the given options:
- A. 50%
- B. 100%
- C. 150%
- D. 200%
Our calculated value of 50% matches Option A.
Revision Table: Key Statistics Concepts
| Concept | Symbol | Definition | Formula (Sample) |
|---|---|---|---|
| Mean | \(\bar{x}\) | Average of a dataset | \(\bar{x} = \frac{\sum x_i}{n}\) |
| Standard Deviation | \(s\) or \(\sigma\) | Measure of spread around the mean | \(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\) |
| Variance | \(s^2\) or \(\sigma^2\) | Average of squared differences from the mean (SD squared) | \(s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\) |
| Coefficient of Variance | CV | Relative measure of variability | \(\text{CV} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100\%\) |
Additional Information on Coefficient of Variance
The Coefficient of Variance is useful in several scenarios:
- Comparing Variability: It is used to compare the degree of variation between two or more datasets, even if their means are drastically different. For example, comparing the variability in height (mean around 170 cm) versus weight (mean around 70 kg) in a population.
- Risk Assessment: In finance, CV can be used to compare the risk (standard deviation) of different investments relative to their expected returns (mean). A lower CV indicates less risk per unit of return.
- Data Quality: In laboratory or industrial settings, CV can be used to assess the consistency and precision of measurements.
It is important to note that the Coefficient of Variance is meaningful only when the mean is a positive, non-zero value. If the mean is zero or negative, the CV is undefined or difficult to interpret.
Important Questions from Standard Deviation
Calculate the mean from the following table.
Scores
Frequencies
0-10
2
10-20
4
20-30
12
30-40
21
40-50
6
50-60
3
60-70
2
Find the standard deviation of the following data (rounded off to two decimal places).
5, 3, 4, 7
If the standard deviation of a population is 5, what will be its variance?
A. 10
B. 15
C. 25
D. 12.5
The variance of a set of data is 196. Then the standard deviation of the data is.
A. ± 14
B. 14
C. 96
D. 98The variance of a set of data is 144. Then the standard deviation of the data is:
A. ±12
B. 12
C. 44
D. 72
