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RD Sharma Class 11 Solutions Chapter 32 - Statistics (Ex 32.4) Exercise 32.4 - Free PDF

RD Sharma Class 11 Solutions Chapter 32 - Statistics (Ex 32.4) Exercise 32.4 - Free PDF

Q: 3. Which key formulas are essential for solving the problems in RD Sharma Class 11 Chapter 32, Exercise 32.4?

Exercise 32.4 primarily deals with variance and standard deviation for grouped data. The most crucial formulas are:Variance (σ²): σ² = (1/N) * Σfᵢ(xᵢ - μ)²Standard Deviation (σ): σ = √[(1/N) * Σfᵢ(xᵢ - μ)²]Here, N is the total frequency (Σfᵢ), xᵢ represents the class mid-point, and μ is the distribution's mean. Understanding the shortcut or step-deviation method variations of these formulas is also vital.

Q: 4. Why is the step-deviation method often preferred for calculating variance in problems with large numbers?

The step-deviation method is preferred because it simplifies complex calculations. When dealing with large values for class mid-points (xᵢ) or frequencies, this method reduces the deviations into smaller, manageable integers by using an assumed mean and a common factor (h). This significantly lowers the chance of calculation errors and makes the process of finding variance and standard deviation much faster and more efficient.

Q: 6. How does variance help in comparing the consistency of two different datasets?

Variance is a measure of the spread or dispersion of data points around the mean. When comparing two datasets, the one with the lower variance is considered more consistent because its data points are clustered more closely together. Conversely, a dataset with a higher variance indicates greater variability and less consistency. This concept is fundamental to analysing frequency distributions in Chapter 32.

Q: 7. In statistics, what is the practical meaning of a standard deviation of zero?

A standard deviation of zero signifies that there is no variability or spread in the data whatsoever. This is a special case where every single data point in the set is identical to the mean. For example, if every student in a class scores exactly 75 marks, the mean is 75 and the standard deviation is 0, indicating perfect consistency.

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Vedantu’s RD Sharma Free PDF For Class 11 Solutions Available

Statistics is the way of summarizing in brief; the way to know and measure. You know a few things about any item or object; like the temperature of the air is 70 degrees Celsius, it can be measured through a thermometer; you can also know about a person; you can know the height of the person, weight, age of the person, income and all that.

From the above definition, we can understand what Statistics is. It is the process of analyzing and recording information to understand how it affects the decisions that we make and how we choose to live our life. It is a science that studies and looks for patterns, to find connections, relationships and trends among things. Statistics can be a bit daunting, so here are some points to help you to understand the subject better.

Statistical Terms and Definitions

Before starting, we will take a look at some basic terms and definitions that are mostly used in statistics and to understand the subject better.

Data: Data can be anything that is collected or observed. It can be information collected from the observations of a particular set of objects. It includes both tangible and intangible things.

Example: The data is the source of information that is collected and analyzed. It is information that shows us about our world. The data we have in our minds, the experiences that we have, the books we read, the conversations that we hear, the things we see, everything is our data.

Records: Records are considered as the list of collected data or information that is recorded on a particular topic or subject. It includes data about all the facts about a particular thing. It is important to know that every record contains several pieces of data.

Example: An internet history is a record of what has happened on the internet.

Variables: Variables are the elements that we will use to measure. It can be an answer that we want to get and the variables that will be used to solve the problem.

Example: The total population of the world is the variable that we will use to find out how many people are left in the world.

Variance: The variance is the variation or the difference in the values of the variable. The value of the difference in one variable and its average. The higher the variance, the higher the variation in that set of values.

Example: When we take the average of 3 numbers then the difference is between 2 and 3. If we multiply the numbers then it is the number in the last row minus the number in the first row.

Standard Deviation: The standard deviation is a measure of the standard of the variation that exists in the data set.

Example: If there are 50 data in our set, then we will use the average of the set as a mean and we will find the standard deviation using the average of the set.

Normal Distribution: It is one of the five common distributions of continuous data. The Normal Distribution has a bell-like shape with most of its data on one side. There are two peaks of the distribution. The peak at the mean is most of the data points. The peak at the mean is not symmetrical. The variance of the distribution is 1.0.

Example: If we want to find out the number of people in the group, then the mean of this group is 100 people and the standard deviation is 10 people. This means that most of the people are concentrated at the right side of the bell-like distribution. The mean will be approximately a value of 60 people.

Percentiles: The percentiles are numerical values that are associated with points in the distribution. These points are chosen to represent the distribution.

Example: There are four numbers which have an average of 15. Suppose that these numbers are 15, 17, 20, and 30. Then these are the percentiles.

InterQuartile Range: The InterQuartile Range is the difference between the 25th percentile and the 75th percentile. It is equal to the 75th percentile - 25th percentile. If the IQR is 0, then the 25th and 75th percentiles are equal.

Example: Given the average of the four numbers is 15 and suppose the 25th percentile is 10 and the 75th percentile is 20, then the IQR is 15 - 10 = 5 and the InterQuartile Range is 75th percentile - 25th percentile = 5. The difference between the 75th percentile and the 25th percentile is 50%.

Variance: The variance is a mathematical measure of how much one measurement varies from the average.

Example: If we assume that the standard deviation of four numbers are 10, 5, 4, and 3, then the variance will be

Skewness: Skewness is the measure of the shape of the distribution.

Example: The skewness of a distribution is equal to (Median - Average) / Standard Deviation. Suppose we have two groups of 30 numbers. One group consists of numbers from 5 to 30 and the other group consists of numbers from 20 to 30. The average of the first group is equal to 15 and the average of the second group is equal to 25. The median of the first group is 15 and the median of the second group is 25. The standard deviation of the first group is equal to 5. The standard deviation of the second group is equal to 2. Then the skewness of the first group of 30 numbers is (15 - 15) / (5) = 0 and the skewness of the second group of 30 numbers is (25 - 15) / (2) = 10. The skewness of the first group of 30 numbers is less than 0 and the skewness of the second group of 30 numbers is greater than 0.

Free PDF download of RD Sharma Class 11 Solutions Chapter 32 - Statistics Exercise 32.4 solved by Expert Mathematics Teachers on Vedantu. All Chapter 32 - Statistics Ex 32.4 Questions with Solutions for RD Sharma Class 11 Maths to help you to revise the complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other engineering entrance exams.

FAQs on RD Sharma Class 11 Solutions Chapter 32 - Statistics (Ex 32.4) Exercise 32.4 - Free PDF

1. Where can I find accurate, step-by-step solutions for RD Sharma Class 11 Maths Exercise 32.4?

Vedantu provides detailed and expert-verified solutions for every problem in RD Sharma Class 11 Maths Chapter 32, Exercise 32.4. These solutions are crafted to explain the correct method for calculating variance and standard deviation, following the latest 2025-26 CBSE curriculum guidelines for clear understanding and exam preparation.

2. What is the correct method to calculate the standard deviation for a continuous frequency distribution in Exercise 32.4?

To calculate the standard deviation (σ) for a continuous frequency distribution, you should follow these steps precisely:

First, determine the mid-point (xᵢ) for each class interval.
Calculate the mean (μ) of the entire distribution using the formula μ = (Σfᵢxᵢ) / N.
Find the deviation of each mid-point from the mean (dᵢ = xᵢ - μ).
Square each of these deviations to get dᵢ².
Multiply each squared deviation by its corresponding frequency (fᵢdᵢ²).
The variance (σ²) is the mean of these values: σ² = (1/N) * Σfᵢdᵢ².
Finally, the standard deviation is the square root of the variance (σ = √σ²).

3. Which key formulas are essential for solving the problems in RD Sharma Class 11 Chapter 32, Exercise 32.4?

Exercise 32.4 primarily deals with variance and standard deviation for grouped data. The most crucial formulas are:

Variance (σ²): σ² = (1/N) * Σfᵢ(xᵢ - μ)²
Standard Deviation (σ): σ = √[(1/N) * Σfᵢ(xᵢ - μ)²]

Here, N is the total frequency (Σfᵢ), xᵢ represents the class mid-point, and μ is the distribution's mean. Understanding the shortcut or step-deviation method variations of these formulas is also vital.

4. Why is the step-deviation method often preferred for calculating variance in problems with large numbers?

The step-deviation method is preferred because it simplifies complex calculations. When dealing with large values for class mid-points (xᵢ) or frequencies, this method reduces the deviations into smaller, manageable integers by using an assumed mean and a common factor (h). This significantly lowers the chance of calculation errors and makes the process of finding variance and standard deviation much faster and more efficient.

5. What is a common mistake to avoid when calculating standard deviation for grouped data?

A frequent error students make is forgetting to multiply the squared deviation, (xᵢ - μ)², by its corresponding frequency (fᵢ). Forgetting this step is equivalent to treating the grouped data as ungrouped, leading to an incorrect result. Always ensure that the frequency of each class is factored into the sum of squared deviations, as the formula is Σfᵢ(xᵢ - μ)².

6. How does variance help in comparing the consistency of two different datasets?

Variance is a measure of the spread or dispersion of data points around the mean. When comparing two datasets, the one with the lower variance is considered more consistent because its data points are clustered more closely together. Conversely, a dataset with a higher variance indicates greater variability and less consistency. This concept is fundamental to analysing frequency distributions in Chapter 32.

7. In statistics, what is the practical meaning of a standard deviation of zero?

A standard deviation of zero signifies that there is no variability or spread in the data whatsoever. This is a special case where every single data point in the set is identical to the mean. For example, if every student in a class scores exactly 75 marks, the mean is 75 and the standard deviation is 0, indicating perfect consistency.