CBSE Class 10 Mathematics
(Chapter 13: Statistics)
20 Important Questions and Answers
As per CBSE Syllabus 2026–27
Q1. What is Statistics? Why is it important in daily life?
Answer:
Statistics is the branch of mathematics that deals with the collection, organization, presentation, analysis, and interpretation of numerical data. It helps us understand large amounts of information in a simple and meaningful way. Statistics is widely used in education, business, sports, economics, healthcare, and government planning. For example, average marks of students, weather reports, population census, and election surveys are based on statistical data. By studying statistics, we can compare different sets of data and make informed decisions. Thus, statistics plays an important role in understanding trends and solving practical problems in everyday life.
Q2. Define mean and explain its significance.
Answer:
The mean, also known as the arithmetic mean, is a measure of central tendency that represents the average value of a data set. It is calculated by dividing the sum of all observations by the total number of observations. The mean helps in understanding the overall performance or trend of a group. It is widely used because it considers every observation in the data. For example, if the marks of students are known, the mean indicates the average performance of the class. However, extreme values may affect the mean significantly. Therefore, while useful, it should sometimes be considered along with other measures like median and mode.
Q3. What is the median? How is it determined?
Answer:
The median is the middle value of a data set when the observations are arranged in ascending or descending order. It divides the data into two equal parts. For an odd number of observations, the median is the middle term. For an even number of observations, it is the average of the two middle terms. In grouped data, the median is calculated using a formula involving the median class, cumulative frequency, class size, and frequency of the median class. The median is useful because it is not affected by extreme values. Therefore, it is often preferred for representing data with unusually high or low observations.
Q4. Define mode and mention its uses.
Answer:
Mode is the value that occurs most frequently in a data set. It represents the observation with the highest frequency. In grouped data, the modal class is the class interval with the highest frequency, and the mode is calculated using a specific formula. Mode is useful in situations where the most common or popular item is required. For example, clothing manufacturers may use mode to determine the most common shirt size demanded by customers. Unlike the mean, mode is not affected by extreme values. It is particularly useful for qualitative data and helps identify the most typical observation in a data set.
Q5. What are the different measures of central tendency?
Answer:
Measures of central tendency are statistical values that represent the central or typical value of a data set. The three main measures are mean, median, and mode. The mean is the arithmetic average of all observations. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently. Each measure has its own advantages. The mean uses all observations, the median is resistant to extreme values, and the mode indicates the most common value. These measures help summarize large data sets and make comparisons easier. They are widely used in statistics and data analysis.
Q6. Explain the direct method for finding the mean of grouped data.
Answer:
The direct method is used to calculate the mean of grouped data by using class marks and frequencies. First, the class mark of each class interval is found by taking the average of its lower and upper limits. Then, each class mark is multiplied by its corresponding frequency. The products are added to obtain the sum of (f_i x_i). Finally, the mean is calculated using the formula:
[\text{Mean}=\frac{\sum f_i x_i}{\sum f_i}]
This method is simple and straightforward when the class intervals and frequencies are manageable. It provides an accurate average by considering all observations represented in the grouped data.
Q7. What is the assumed mean method?
Answer:
The assumed mean method is a simplified technique for calculating the mean of grouped data. Instead of using actual class marks directly, one class mark is chosen as the assumed mean. Deviations of other class marks from the assumed mean are calculated. These deviations are multiplied by their respective frequencies and summed. The mean is then obtained using the formula:
[\bar{x}=A+\frac{\sum f_i d_i}{\sum f_i}]
where (A) is the assumed mean and (d_i) is the deviation. This method reduces calculations and is especially useful when class marks are large numbers. It saves time and minimizes computational errors.
Q8. What is the step-deviation method?
Answer:
The step-deviation method is an advanced form of the assumed mean method used for grouped data with equal class intervals. In this method, deviations from the assumed mean are divided by the common class width to simplify calculations. The formula used is:
[\bar{x}=A+\left(\frac{\sum f_i u_i}{\sum f_i}\right)h]
where (A) is the assumed mean, (u_i) is the step deviation, and (h) is the class size. This method makes calculations easier when the values involved are large. It is widely used in statistics because it reduces arithmetic work while giving the same result as the direct and assumed mean methods.
Q9. What is a cumulative frequency?
Answer:
Cumulative frequency is the running total of frequencies in a frequency distribution. It is obtained by successively adding frequencies from the first class interval to the last. Cumulative frequencies help determine the median and construct cumulative frequency curves, also known as ogives. There are two types: less-than cumulative frequency and more-than cumulative frequency. Less-than cumulative frequency is calculated by adding frequencies progressively, while more-than cumulative frequency is found by subtracting frequencies from the total. Cumulative frequencies help understand how observations are distributed and make it easier to locate specific positions within a data set.
Q10. What is a median class?
Answer:
The median class is the class interval in a grouped frequency distribution that contains the median value. To identify it, cumulative frequencies are calculated, and the value (\frac{N}{2}) is determined, where (N) is the total frequency. The class whose cumulative frequency first exceeds (\frac{N}{2}) is called the median class. This class is used in the median formula to calculate the exact median value. The median class plays an important role because it helps locate the central position of grouped data accurately. Without identifying the median class, the median of grouped data cannot be determined correctly.
Q11. State the formula for finding the median of grouped data.
Answer:
The median of grouped data is calculated using the formula:
[\text{Median}=l+\left(\frac{\frac{N}{2}-cf}{f}\right)h]
where (l) is the lower boundary of the median class, (N) is the total frequency, (cf) is the cumulative frequency before the median class, (f) is the frequency of the median class, and (h) is the class size. This formula helps determine the central value of grouped data accurately. The median divides the distribution into two equal parts. It is particularly useful when the data contains extreme values, as it is less affected by them than the mean.
Q12. What is a modal class?
Answer:
A modal class is the class interval in a grouped frequency distribution that has the highest frequency. Since the mode represents the most frequently occurring value, the class with the maximum frequency is selected as the modal class. After identifying the modal class, the mode is calculated using the mode formula. The modal class provides information about the concentration of observations in a particular interval. It helps identify the most common range of values in a data set. Modal classes are useful in practical situations such as determining popular product sizes, preferred age groups, or frequently occurring scores.
Q13. Write the formula for mode in grouped data.
Answer:
The mode of grouped data is calculated using the formula:
[\text{Mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)h]
where (l) is the lower boundary of the modal class, (f_1) is the frequency of the modal class, (f_0) is the frequency of the class preceding the modal class, (f_2) is the frequency of the class succeeding the modal class, and (h) is the class width. This formula helps estimate the most frequently occurring value in grouped data. Mode is useful because it identifies the value around which observations are most concentrated.
Q14. Why is the median preferred over the mean in some cases?
Answer:
The median is preferred over the mean when the data contains extreme values or outliers. Since the median depends only on the position of observations and not on their actual values, it remains unaffected by unusually high or low observations. For example, in income distributions where a few people earn exceptionally high salaries, the mean may give a misleading picture of average income. In such cases, the median provides a more realistic representation of the central tendency. Therefore, the median is widely used in economics, social sciences, and other fields where data may be skewed or unevenly distributed.
Q15. How does mode differ from mean and median?
Answer:
Mode differs from mean and median because it identifies the most frequently occurring value rather than the average or middle value. The mean is calculated using all observations, while the median depends on the position of observations. Mode focuses only on frequency. Unlike the mean, mode is not affected by extreme values. It is especially useful for categorical or qualitative data where numerical averages may not be meaningful. For example, the most common shoe size sold in a store is represented by the mode. Thus, mode highlights popularity, whereas mean and median describe the center of the data.
Q16. What are class intervals and class marks?
Answer:
A class interval is a range of values used to group data in a frequency distribution. It consists of a lower limit and an upper limit. For example, 10–20 is a class interval. The class mark is the midpoint of a class interval and is calculated by averaging its lower and upper limits. For the interval 10–20, the class mark is:
[\frac{10+20}{2}=15]
Class marks are used in calculating the mean of grouped data. They represent all observations within a class interval. Thus, class intervals organize data, while class marks help simplify statistical calculations.
Q17. What is meant by grouped data?
Answer:
Grouped data refers to data that has been organized into class intervals along with their corresponding frequencies. Instead of listing individual observations, similar values are grouped together. This method is useful when dealing with a large number of observations. Grouped data makes it easier to understand patterns, trends, and distributions. Statistical measures such as mean, median, and mode can be calculated efficiently from grouped data. For example, students’ marks can be arranged in intervals like 0–10, 10–20, and so on. Grouping data reduces complexity and provides a clearer representation of large datasets.
Q18. Explain the relationship between mean, median, and mode.
Answer:
For a moderately skewed distribution, there is an empirical relationship between mean, median, and mode:
[\text{Mode}=3(\text{Median})-2(\text{Mean})]
This formula helps estimate one measure when the other two are known. Although it is not exact for every distribution, it provides a useful approximation. The relationship shows how different measures of central tendency are connected. In a perfectly symmetrical distribution, the mean, median, and mode are equal. Understanding this relationship helps in checking calculations and analyzing data more effectively. It is an important concept in statistics and frequently appears in CBSE examinations.
Q19. What are the advantages of using grouped frequency distributions?
Answer:
Grouped frequency distributions help organize large amounts of data into manageable class intervals. They simplify the presentation and interpretation of information. By grouping data, patterns and trends become easier to identify. Calculations of statistical measures such as mean, median, and mode become more systematic. Grouped distributions also reduce the complexity of raw data and make comparisons easier. They are widely used in schools, surveys, research studies, and government reports. Although some detailed information may be lost, grouped frequency distributions provide a clear and concise summary of large datasets, making statistical analysis more efficient.
Q20. Why are measures of central tendency important in statistics?
Answer:
Measures of central tendency are important because they summarize a large set of data into a single representative value. They help understand the general behavior and characteristics of a dataset. Mean, median, and mode make comparisons between different groups easier and provide meaningful insights into trends and patterns. These measures are widely used in education, economics, business, sports, and scientific research. They assist in decision-making and data interpretation. Without measures of central tendency, analyzing large datasets would be difficult and time-consuming. Therefore, they form the foundation of statistical analysis and data representation.