Question 1 of 21intermediate🔧 ApplyNumerical4 marks

A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

Number of plants0 - 22 - 44 - 66 - 88 - 1010 - 1212 - 14
Number of houses1215623

Which method did you use for finding the mean, and why?

Correct Answer

To find the mean number of plants per house, we use the Direct Method.

First, we calculate the class mark (xix_i) for each interval, which is the midpoint of the class. Then, we compute the product of frequency and class mark (fixif_i x_i).

Number of plantsNumber of houses (fif_i)Class mark (xix_i)fixif_i x_i
0 - 2111
2 - 4236
4 - 6155
6 - 85735
8 - 106954
10 - 1221122
12 - 1431339
TotalΣfi=20Σ f_i = 20Σfixi=162Σ f_i x_i = 162

Mean =∑fixi∑fi=16220=8.1= \frac{\sum f_i x_i}{\sum f_i} = \frac{162}{20} = 8.1

Method Used: I used the Direct Method because the class sizes were small and the values of the class marks (xix_i) were small integers. This made the calculation of fixif_i x_i simple and manageable without needing to assume a mean or use the step-deviation method.

Exercise: EXERCISE 13.1 | Q: 1 | (Page 11)
For More Understanding

Explanation

The question requires finding the mean from a grouped frequency distribution. The Direct Method is chosen because the numerical values involved (class marks and frequencies) are small integers, making direct computation straightforward and less prone to calculation errors compared to methods involving assumed means. The solution involves finding the midpoint (class mark) for each interval, multiplying it by the frequency, summing these products, and dividing by the total frequency.

Solution Steps

  1. Step 1: Identify the class intervals and frequencies (fif_i) from the given data.

  2. Step 2: Calculate the class mark (xix_i) for each interval by finding the midpoint (e.g., for 0-2, xi=(0+2)/2=1x_i = (0+2)/2 = 1).

  3. Step 3: Calculate the product fixif_i x_i for each row (e.g., for the first row, 1×1=11 \times 1 = 1).

  4. Step 4: Sum the frequencies to get ∑fi=20\sum f_i = 20 and sum the products to get ∑fixi=162\sum f_i x_i = 162.

  5. Step 5: Apply the Direct Method formula: Mean =∑fixi∑fi=16220=8.1= \frac{\sum f_i x_i}{\sum f_i} = \frac{162}{20} = 8.1.