Consider the following statements: 1. Variance is unaffected by change of origin and change of scale. 2. Coefficient of variance is independent of the unit of observations.
2 only
Let's analyze the given statements regarding the properties of variance and the coefficient of variance. We will consider how these statistical measures are affected by transformations of the data, specifically changes in origin, changes in scale, and the units of measurement.
Analyzing Properties of Variance and Coefficient of Variance
Statement 1: Variance is unaffected by change of origin and change of scale.
Let $X$ be a variable with mean $\mu$ and variance $\sigma^2 = \text{Var}(X)$.
Change of Origin: A change of origin means adding a constant value to each observation. Let $Y = X + c$, where $c$ is a constant.
The mean of $Y$ is $E(Y) = E(X + c) = E(X) + c = \mu + c$.
The variance of $Y$ is $\text{Var}(Y) = E[(Y - E(Y))^2] = E[((X + c) - (\mu + c))^2] = E[(X - \mu)^2] = \text{Var}(X) = \sigma^2$.
This shows that variance is indeed unaffected by a change of origin.
Change of Scale: A change of scale means multiplying each observation by a constant value. Let $Z = bX$, where $b$ is a constant.
The mean of $Z$ is $E(Z) = E(bX) = b E(X) = b\mu$.
The variance of $Z$ is $\text{Var}(Z) = E[(Z - E(Z))^2] = E[(bX - b\mu)^2] = E[b^2(X - \mu)^2] = b^2 E[(X - \mu)^2] = b^2 \text{Var}(X) = b^2\sigma^2$.
This shows that variance *is* affected by a change of scale; it is multiplied by the square of the scale factor.
If we combine both transformations, $W = a + bX$, then $\text{Var}(W) = \text{Var}(a + bX) = b^2\text{Var}(X) = b^2\sigma^2$. Variance is unaffected by the additive constant ($a$, change of origin) but is affected by the multiplicative constant ($b$, change of scale).
Therefore, the statement "Variance is unaffected by change of origin and change of scale" is false because variance is affected by a change of scale.
Statement 2: Coefficient of variance is independent of the unit of observations.
The Coefficient of Variance (CV) is defined as the ratio of the standard deviation ($\sigma$) to the mean ($\mu$), usually expressed as a percentage:
$\text{CV} = \left(\frac{\sigma}{\mu}\right) \times 100\%$
Standard deviation ($\sigma$) has the same units as the original observations and the mean ($\mu$). For example, if the observations are in meters, both $\sigma$ and $\mu$ are in meters.
When calculating the ratio $\frac{\sigma}{\mu}$, the units in the numerator and the denominator cancel out. For instance, $\frac{\text{meters}}{\text{meters}}$ results in a dimensionless quantity. Multiplying by 100% just expresses this ratio as a percentage, which is also unitless.
Because the CV is a ratio of two quantities with the same units, the units cancel out, making the CV a relative measure of dispersion that is independent of the units of measurement used for the observations.
Therefore, the statement "Coefficient of variance is independent of the unit of observations" is true.
Conclusion on Correct Statements
Based on our analysis:
- Statement 1 is false (Variance is affected by change of scale).
- Statement 2 is true (Coefficient of variance is independent of the unit of observations).
Thus, only Statement 2 is correct.
| Statistic | Affected by Change of Origin (adding a constant $a$) | Affected by Change of Scale (multiplying by a constant $b$) | Independent of Units |
|---|---|---|---|
| Variance ($\sigma^2$) | No (Var($X+a$) = Var($X$)) | Yes (Var($bX$) = $b^2$Var($X$)) | No (units are squared) |
| Standard Deviation ($\sigma$) | No (SD($X+a$) = SD($X$)) | Yes (SD($bX$) = $|b|$SD($X$)) | No (same units as data) |
| Mean ($\mu$) | Yes ($E(X+a) = E(X) + a$) | Yes ($E(bX) = bE(X)$) | No (same units as data) |
| Coefficient of Variance (CV) | No (if $a+\mu \ne 0$) | No (CV($bX$) = CV($X$)) | Yes |
Revision Table: Statistical Measures Properties
| Statistic | Original ($X$) | Transformed ($Y = a + bX$) |
|---|---|---|
| Mean | $\mu_X$ | $\mu_Y = a + b\mu_X$ |
| Variance | $\sigma_X^2$ | $\sigma_Y^2 = b^2\sigma_X^2$ |
| Standard Deviation | $\sigma_X$ | $\sigma_Y = |b|\sigma_X$ |
Additional Information: Understanding Coefficient of Variance
The Coefficient of Variance (CV) is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage. Its main advantages are:
- Comparison across different scales: Since it's unitless, the CV is useful for comparing the degree of variation between data sets, even if their means are drastically different or if they are measured in different units (e.g., comparing the variation in height measured in centimeters to the variation in weight measured in kilograms).
- Relative variability: It provides a measure of relative variability rather than absolute variability (like standard deviation). This is helpful when comparing the variability of variables with different means. A standard deviation of 10 might be large if the mean is 20, but small if the mean is 1000. The CV accounts for this difference in scale.
A higher CV indicates higher variability relative to the mean, while a lower CV indicates lower variability relative to the mean.
Important Questions from Variance and Standard Deviation
The sum of deviations of n numbers from 10 and 20 are p and q respectively. If (p - q)2 = 10000, then what is the value of n?
The mean and the variance of 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively.
If the data are moderately non-symmetrical, then which one of the following empirical relationships is correct?
If the total number of observations is 20, ∑ x i= 1000 and \(\sum {\rm{x}}_{\rm{i}}^2 = 84000\) , then what is the variance of the distribution?
Among these options, which one is NOT an example of relative measure of dispersion?
