NCERT Solutions Class 9 Maths Chapter 14 Exercise 14.4

Measures of Central Tendency: Mean, Median, and Mode

In statistics, a single value that represents the center or midpoint of a data distribution is called a measure of central tendency. In Exercise 14.4 of Class 9 CBSE Mathematics, we focus on three major measures:

1. Mean (Arithmetic Average): Calculated by summing all observations and dividing by the total count of observations. Formula:

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

1. Median (Middle Value): The middlemost observation when the data is sorted in ascending order. It splits the data into two equal halves. If the total number of observations $n$ is odd, Median = $(\frac{n+1}{2})^{\text{th}}$ term. If $n$ is even, Median = Average of $(\frac{n}{2})^{\text{th}}$ and $(\frac{n}{2} + 1)^{\text{th}}$ terms.
2. Mode (Most Frequent Value): The observation that occurs most frequently in the dataset. A dataset can have one mode, multiple modes, or no mode at all.

Understanding when to use each measure is key. The Mean is sensitive to extreme values (outliers), whereas the Median provides a more robust representation of central tendency in skewed data distributions.

Step-by-Step Process to Calculate the Median

1. Sort the Dataset — Arrange all given observations in ascending (or descending) order. This is a crucial step; calculating the median on raw, unsorted data is the most common student error.
2. Count the Number of Observations (n) — Count how many items are in the dataset. Let this total number of observations be $n$.
3. Check if n is Odd or Even — If $n$ is odd, use the formula: $\text{Median} = \text{Value of } \left(\frac{n+1}{2}\right)^{\text{th}} \text{ term}$. If $n$ is even, use: $\text{Median} = \frac{\left(\text{Value of } \left(\frac{n}{2}\right)^{\text{th}} \text{ term}\right) + \left(\text{Value of } \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ term}\right)}{2}$.

Solved NCERT Style Examples

• Example 1: Find the mean, median, and mode of the following marks obtained by 9 students: 41, 39, 48, 52, 46, 62, 54, 40, 96. Solution: - Mean: Sum of observations = 41 + 39 + 48 + 52 + 46 + 62 + 54 + 40 + 96 = 478. Number of observations ($n$) = 9. Mean = 478 / 9 = 53.11. - Median: First, arrange data in ascending order: 39, 40, 41, 46, 48, 52, 54, 62, 96. Since $n = 9$ (odd), Median = $(\frac{9+1}{2})^{\text{th}}$ observation = $5^{\text{th}}$ observation. The $5^{\text{th}}$ term is 48. Thus, Median = 48. - Mode: All values appear exactly once. Hence, there is no single distinct mode.
• Example 2: The points scored by a basketball team in a series of 10 matches are: 17, 2, 7, 27, 25, 5, 14, 18, 10, 24. Find the median of this data. Solution: - Arrange the data in ascending order: 2, 5, 7, 10, 14, 17, 18, 24, 25, 27. - Number of observations ($n$) = 10 (even). - Formula: Median = Average of $(\frac{n}{2})^{\text{th}}$ and $(\frac{n}{2} + 1)^{\text{th}}$ observations. - $\frac{10}{2} = 5^{\text{th}}$ term = 14. - $\frac{10}{2} + 1 = 6^{\text{th}}$ term = 17. - Median = $\frac{14 + 17}{2} = \frac{31}{2} = 15.5$.

Important Exam Tips & Common Mistakes

• Always Sort First: The single most common mistake students make in the CBSE Class 9 exam is calculating the median directly from raw, unsorted data. Always sort in ascending order first!
• Calculating the Even Median Terms: When $n$ is even, remember to find the average of the two middle terms. Do not just take the term at index $n/2$.
• Calculation of Mean: Ensure your summation is 100% correct. Re-add the numbers from bottom to top to double-check your calculations on your rough sheet.

Practice Questions with Solutions

• Q: Find the mean of the first five prime numbers. A: Step 1: Write down the first five prime numbers. They are 2, 3, 5, 7, and 11. Step 2: Use the mean formula: Mean = (Sum of all observations) / (Total number of observations). Step 3: Sum = 2 + 3 + 5 + 7 + 11 = 28. Step 4: Count of numbers (n) = 5. Step 5: Calculate Mean = 28 / 5 = 5.6. Final answer: The mean of the first five prime numbers is 5.6.
• Q: The following observations have been arranged in ascending order: 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95. If the median of the data is 63, find the value of x. A: Step 1: Note that the data is already sorted in ascending order. Step 2: Count the number of observations, n = 10 (which is even). Step 3: Since n is even, Median = Average of (n/2)th and (n/2 + 1)th observations. Step 4: Here, the 5th term is x, and the 6th term is x + 2. Step 5: Median = [x + (x + 2)] / 2 = (2x + 2) / 2 = x + 1. Step 6: We are given Median = 63. So, x + 1 = 63 => x = 62. Final answer: The value of x is 62.
• Q: Find the mode of the following data: 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18. A: Step 1: Organize the data by writing the frequency of each observation. Step 2: Frequencies: 14 appears 4 times; 18 appears 3 times; 25, 28, 17, 23, 22 each appear 1 time. Step 3: Identify the observation with the maximum frequency. 14 has the highest frequency (4 times). Final answer: The mode of the given data is 14.
• Q: Calculate the median of the following dataset: 45, 34, 56, 12, 89, 67, 21. A: Step 1: Arrange the data in ascending order: 12, 21, 34, 45, 56, 67, 89. Step 2: Count the number of observations (n). Here, n = 7 (which is odd). Step 3: Apply the median formula for odd n: Median = ((n + 1)/2)th term. Step 4: Median = ((7 + 1)/2)th term = 4th term. Step 5: The 4th term in the sorted list is 45. Final answer: The median is 45.

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