Practice Questions

Analyze the class interval $22.5 - 37.5$ and calculate its class mark.

State the formula for calculating the mean of grouped data using the Direct Method.

Calculate the class size and the class mark for the class interval $100-120$.

Explain the meaning of each symbol in the formula for calculating the mode of grouped data: $\text{Mode} = l + \left(\frac{f_{1} - f_{0}}{2f_{1} - f_{0} - f_{2}}\right) \times h$

Identify the term used for the class interval with the highest frequency in a grouped data distribution.

Propose a modification to the class intervals of the data set 50-59, 60-69, 70-79 to make it suitable for median calculation and justify the necessity of this change.

Define the term 'class mark' as used in the context of grouped data.

Examine the following frequency distribution and identify the modal class.

A shoe company wants to decide which shoe size to produce the most. Justify which measure of central tendency (mean, median, or mode) is the most appropriate for this decision.

The distribution below shows the number of runs scored by batsmen in a cricket season. Calculate the mode of the runs scored.

The weights of 60 students in a class are given in the following distribution. Calculate the median weight.

The following distribution gives the daily expenditure on milk of 30 households in a locality. Calculate the mean daily expenditure by using the assumed mean method.

The following table gives the production yield per hectare of wheat of 100 farms of a village. Calculate the mean production yield using the step-deviation method.

Describe the three different methods for calculating the mean of grouped data. For each method, state its name, its formula, and explain briefly when it is most appropriate to use.

Recall and state the empirical relationship between the three measures of central tendency: mean, median, and mode.

What is a 'cumulative frequency distribution'?

Explain the purpose of choosing an 'assumed mean' in the Assumed Mean Method for calculating the mean of grouped data.

Consider the following frequency distribution table. From this table, identify: (a) The modal class (b) The median class

Summarize the Step-deviation method for finding the mean of grouped data by listing the key steps and the final formula.

Summarize the complete procedure for calculating the median of a grouped frequency distribution. List all the steps from beginning to end.

For a frequency distribution, the sum of frequencies is $N=60$. Analyze the cumulative frequency table below and determine the median class.

For a moderately skewed distribution, the mean is calculated to be $28.5$ and the median is $30$. Apply the empirical relationship between mean, median, and mode to calculate the mode.

A student calculates the mean of a grouped data set and finds it is lower than the lowest class limit. Critique this result. Is it possible? Justify your answer.

To calculate the mean using the assumed mean method, a student proposes choosing the assumed mean 'a' as 0. Formulate an argument explaining if this is a valid choice and evaluate its usefulness.

The class marks ($x_i$) of a distribution are 125, 175, 225, 275, 325, 375 and their corresponding frequencies ($f_i$) are 20, 15, 25, 30, 12, 8. Justify which method (Direct, Assumed Mean, or Step-Deviation) is most suitable for finding the mean and explain why it is more efficient than the others in this case. You do not need to calculate the mean.

For the given frequency distribution, a student claims the median is 45 because the middle value of the class intervals is 40-50. Evaluate this claim. If it is incorrect, identify the median class and justify your choice.

A dataset has a calculated Mean of 25 and a Mode of 20. A student uses the empirical formula $3 \text{ Median} = \text{Mode} + 2 \text{ Mean}$ to find the median. Critique the reliability of this method. Calculate the median using this formula and explain why it is considered an approximation and not an exact value.

Design a plan to determine the average time (in minutes) students in your school spend on social media daily. Your plan should propose the data collection method, the structure of the grouped frequency table you would use (including class intervals), and justify which measure of central tendency would be most appropriate to report your findings.

The monthly incomes of employees in two startups, 'Innovate Inc.' and 'Tech Forward', are given below.

Innovate Inc.: Mean Income = ₹70,000; Median Income = ₹45,000; Modal Income = ₹40,000.

Tech Forward: Mean Income = ₹50,000; Median Income = ₹48,000; Modal Income = ₹52,000.

A financial analyst claims that 'Innovate Inc.' is a better company to work for because its mean income is much higher. Critique this claim. Justify which company likely offers a more equitable salary structure for a typical employee, using the given measures of central tendency to support your argument.

Describe the first two steps to convert a 'less than' type cumulative frequency distribution into a normal frequency distribution.

The following table shows the daily income of 50 families in a locality. Calculate the mean daily income using the direct method.

The formula for the mode of a grouped data is $\text{Mode} = l+\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) \times h$. Formulate a real-world scenario represented by a grouped frequency distribution where the modal class is 40-50, but the calculated mode is less than 45. Justify your formulation with appropriate values for $l, f_1, f_0, f_2,$ and $h$.

Describe what the 'mean' and 'median' of a data set represent and explain the main difference in how they are affected by extreme values (outliers).

Consider the following data for the weights of students:

If 5 students from the '40-45' kg class were mistakenly recorded and their actual weight is in the '60-65' kg class, evaluate how this correction would impact the mean and the modal class. Justify your answer without calculating the exact new mean.

The data regarding the heights of 50 girls of Class X is given below. Analyze the data to calculate the median, mean and mode for the given distribution.

The lengths of 40 leaves of a plant are measured correct to the nearest millimeter. The data obtained is represented in the table below. The class intervals are discontinuous. First, convert the distribution to a continuous frequency distribution. Then, calculate the median length of the leaves.

The median of the distribution given below is $46$. The total frequency is 230. Analyze the data to find the values of the missing frequencies $x$ and $y$.

The median of the distribution below is 35. The total number of observations is 170. Create the complete frequency distribution by finding the missing frequencies $f_1$ and $f_2$. After finding the frequencies, evaluate the modal class and calculate the mode for the completed data.

Explain why it is necessary for class intervals to be continuous when calculating the median or mode of grouped data.

For a grouped frequency distribution with unequal class sizes, evaluate the statement: "The step-deviation method cannot be used." Is this statement always true? Justify.

The mean of the following frequency distribution is $25$. Analyze the data to find the value of the missing frequency $p$.

Create a discrete frequency distribution with 5 distinct integer values and a total frequency of 10, such that the mean is exactly 5 and the median is 4. Justify your created distribution.

Formulate a grouped frequency distribution with 5 classes and a total frequency of 50 that is skewed to the right. Create the distribution table. Then, calculate the mean, median, and mode for your created data. Justify that the relationship Mean > Median > Mode holds true for your distribution, and explain why this relationship is characteristic of a right-skewed distribution.

Explain the concept of 'measures of central tendency'. Describe the three main measures (mean, median, and mode) and provide a real-life example for each where it would be the most suitable measure to use.