Practice Questions

Explain why the sum of absolute deviations is taken when calculating mean deviation, instead of the sum of the deviations themselves.

Justify why the sum of deviations of a set of observations from their arithmetic mean is always zero, and explain why this property makes it an unsuitable basis for a measure of dispersion.

The sum of squares of deviations for $10$ observations taken from their mean $50$ is $250$. Calculate the variance.

If the variance of a data set is $64$, calculate its standard deviation.

Explain what is meant by a 'discrete frequency distribution' and a 'continuous frequency distribution'.

Define 'Range' as a measure of dispersion.

State the formula for the variance ($\sigma^2$) of ungrouped data with $n$ observations $x_1, x_2, \ldots, x_n$ and mean $\bar{x}$.

What is the relationship between Variance and Standard Deviation?

Calculate the range for the following set of observations: $18, 25, 12, 45, 21, 52, 19, 33$.

Name the four measures of dispersion mentioned in your chapter.

Two batsmen have the same average score of 55 runs. The standard deviation of scores for Batsman A is 8 runs, while for Batsman B it is 20 runs. Evaluate which batsman is a more dependable choice for a team and justify your answer.

A teacher has the marks of two students, Alex and Ben, from 5 tests: Alex: 85, 90, 82, 88, 95 Ben: 95, 92, 98, 60, 95 The teacher wants to award the "Most Consistent Performer" prize. Propose a statistical method to determine the winner. Justify your choice of method and use it to decide who should get the prize.

Calculate the mean deviation about the mean for the first five prime numbers.

Calculate the mean and variance for the following continuous frequency distribution:

Recall the value of the sum of deviations of a set of observations from their arithmetic mean.

List the steps required to calculate the mean deviation about the mean for a discrete frequency distribution.

Describe the primary difference between a measure of central tendency and a measure of dispersion. Use the example of the two batsmen A and B from the source text to illustrate.

Summarize the procedure for finding the median of a continuous frequency distribution.

Identify the formula for calculating the mean of a discrete frequency distribution with $n$ distinct values $x_1, x_2, \ldots, x_n$ occurring with frequencies $f_1, f_2, \ldots, f_n$. Explain each term in the formula.

Explain why the sum of squares of deviations from the mean, $\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$, is not considered a proper measure of dispersion on its own, and how 'variance' overcomes this issue.

Analyze the effect on the variance of a data set if each observation is decreased by $10$.

Calculate the mean deviation about the median for the following data: $15, 22, 18, 25, 12, 10$.

Calculate the variance and standard deviation for the data: $5, 8, 10, 12, 15$.

Calculate the mean deviation about the mean for the following frequency distribution:

Using the shortcut method, calculate the mean and standard deviation for the following distribution:

Formulate a rigorous algebraic proof to show that variance is independent of the change of origin. That is, if a new set of observations $y_i$ is created by adding a constant 'a' to each observation $x_i$ (i.e., $y_i = x_i + a$), prove that the variance of $y$ is equal to the variance of $x$.

A student claims that if the range of a dataset is zero, its standard deviation must also be zero. Critique this statement. Is it always true? Justify your reasoning.

Create a dataset of 5 distinct integers such that its mean is 8 and its range is 10. Then, calculate the mean deviation about the mean for the dataset you created.

Formulate a dataset of four distinct positive integers whose arithmetic mean is 10 and whose median is 9.

The mean salary in Company A is ₹60,000 with a standard deviation of ₹6,000. In Company B, the mean salary is ₹75,000 with a standard deviation of ₹9,000. Evaluate which company has a more uniform salary structure and justify your answer.

The sum of the squares of the deviations of a set of 10 observations from their mean of 20 is 250. Justify that the standard deviation is 5. Now, if each observation is increased by 5, justify how the new standard deviation is affected.

Describe in detail the steps to calculate the mean deviation about the median for ungrouped data, considering both cases where the number of observations is even and odd.

The mean and standard deviation of 100 observations were calculated as 20 and 3, respectively. It was later discovered that three observations, 21, 21, and 18, were incorrect. If these incorrect observations are removed from the data, critique the expected impact and then formulate the steps to calculate the new mean and new standard deviation for the remaining 97 observations.

If a constant value 'a' is added to each observation in a data set, what is the effect on the variance? Explain why.

Explain the shortcut method for calculating variance for a continuous frequency distribution. Define each term used in the formula $\sigma^{2}=\frac{h^{2}}{\mathrm{~N}^{2}}\left[\mathrm{~N} \sum f_{i} y_{i}^{2}-\left(\sum f_{i} y_{i}\right)^{2}\right]$.

Design a discrete frequency distribution with at least 4 distinct values ($x_i$) and a total frequency ($N$) of 20, such that the mean of the distribution is exactly 10.

Calculate the mean deviation about the median for the following data:

Create two different sets of data, A and B, each with 5 data points. Both sets must have the same mean of 20 and the same range of 16. Then, calculate the standard deviation for both sets. Evaluate which dataset is more dispersed and justify why the range can be a misleading measure of dispersion.

A researcher analyzed the dataset {5, 10, 15, 20, 200} and concluded that since the mean deviation about the mean is relatively small compared to the maximum value, the data has low dispersion. Critique this conclusion. Propose a more appropriate measure of central tendency and dispersion for this specific dataset and justify your choice.

The scores of two batsmen, P and Q, in five innings are given below. Analyze their scores to determine who is the more consistent player. Batsman P: $25, 50, 45, 30, 70$ Batsman Q: $40, 48, 50, 52, 55$

The mean of 5 observations is $6$ and the variance is $6.8$. If three of the observations are $2, 4,$ and $8$, find the other two observations.

Two distributions have the same mean. The first distribution has $10$ observations with a standard deviation of $3$, and the second has $15$ observations with a standard deviation of $2$. If the observations are combined, will the standard deviation of the combined group be less than, equal to, or greater than the average of the two standard deviations? Justify your answer without calculation.

Derive the shortcut formula for variance, $\sigma^2 = \frac{h^2}{N^2} [N \sum f_i y_i^2 - (\sum f_i y_i)^2]$, where $y_i = \frac{x_i - A}{h}$. Justify each step of the derivation starting from the basic definition of variance, $\sigma_x^2 = \frac{1}{N} \sum f_i (x_i - \bar{x})^2$.

The mean and standard deviation of 50 observations were calculated as 60 and 5 respectively. It was later discovered that one observation, 75, was misread as 25. Calculate the correct mean and correct standard deviation.

The variance of a set of observations is 25. If each observation is multiplied by -3, justify what the new variance will be without performing detailed calculations.