GSEB Class 11 Statistics Solutions Chapter 3 Measures of Central Tendency Exercise 3.3

Exercise 3.3

 

Question 1. The following data show the number of books read by 8 students of a class during last month: 2, 1, 5, 9, 1, 3, 2, 4. Find the average number of books read using geometric mean.

Answer: For this dataset, the given values are \( x_1 = 2 \), \( x_2 = 1 \), \( x_3 = 5 \), \( x_4 = 9 \), \( x_5 = 1 \), \( x_6 = 3 \), \( x_7 = 2 \), and \( x_8 = 8 \). The geometric mean (G) of these eight observations is calculated as follows:
\[ G = \sqrt[8]{x_1 \times x_2 \times x_3 \times x_4 \times x_5 \times x_6 \times x_7 \times x_8} \]
\[ G = \sqrt[8]{2 \times 1 \times 5 \times 9 \times 1 \times 3 \times 2 \times 4} \]
\[ G = \sqrt[8]{2160} \] To determine the eighth root of 2160, we apply the square root operation three times successively:
First square root: \( \sqrt{2160} \approx 46.4758 \)
Second square root: \( \sqrt{46.4758} \approx 6.8173 \)
Third square root: \( \sqrt{6.8173} \approx 2.611 \) Therefore, the geometric mean of the number of books read is approximately 2.61 books.
In simple words: To find the average number of books using the geometric mean, we multiply all the book counts and then take the 8th root of that product, which gives us approximately 2.61 books.

🎯 Exam Tip: Remember to calculate the nth root by taking successive square roots if n is a power of 2 (e.g., 4th root is two square roots, 8th root is three square roots). Clearly show the steps for calculating the root for full marks.

 

Question 2. The value of a machine depreciates at the rate of 10%, 7%, 5% and 2% in its first four years respectively. Find the average rate of depreciation using an appropriate method.

Answer: Given that the depreciation rates for a machine are provided as percentages, the most suitable method for finding the average depreciation rate is the geometric mean. First, we determine the remaining value of the machine after each year's depreciation:
\( x_1 = 100 - 10 = 90 \)
\( x_2 = 100 - 7 = 93 \)
\( x_3 = 100 - 5 = 95 \)
\( x_4 = 100 - 2 = 98 \) The geometric mean (G) of these values is calculated as follows:
\[ G = \sqrt[4]{x_1 \times x_2 \times x_3 \times x_4} \]
\[ G = \sqrt[4]{90 \times 93 \times 95 \times 98} \]
\[ G = \sqrt[4]{77924700} \] To compute the fourth root of 77924700, we perform two consecutive square root operations:
First square root: \( \sqrt{77924700} \approx 8827.4965 \)
Second square root: \( \sqrt{8827.4965} \approx 93.95 \) Thus, the geometric mean (average value retention) is 93.95. The average rate of depreciation over these four years is then calculated as \( (100 - 93.95) = 6.05\% \).
In simple words: To find the average depreciation rate, we first calculate the percentage of value retained each year (e.g., 100%-10% = 90%). Then, we find the geometric mean of these retained percentages. Finally, subtract this average retained percentage from 100% to get the average depreciation rate.

🎯 Exam Tip: When dealing with percentage changes or rates (like depreciation or growth), the geometric mean is generally the most appropriate measure of central tendency. Ensure the initial values are correctly converted to factors (e.g., 100 - rate).

 

Question 3. A taxi travelled 15 km on Monday and 254 km on Tuesday. Find the average distance travelled over these two days using geometric mean.

Answer: The distances traveled by the taxi are given as \( x_1 = 15 \) km on Monday and \( x_2 = 254 \) km on Tuesday. To find the average distance using the geometric mean:
\[ G = \sqrt{x_1 \times x_2} \]
\[ G = \sqrt{15 \times 254} \]
\[ G = \sqrt{3810} \]
\[ G \approx 61.725 \] Rounding to two decimal places, the average distance traveled by the taxi is approximately 61.73 km.
In simple words: To calculate the average distance using the geometric mean, we multiply the distances traveled on both days and then take the square root of that product. This gives an average distance of about 61.73 km.

🎯 Exam Tip: The geometric mean is useful when calculating average rates or ratios, or when data values are widely varied, as it mitigates the impact of extreme values. Clearly show the multiplication and square root steps.