The table below gives the percentage distribution of female teachers in the primary schools of rural areas of various states and union territories (U.T.) of India. Find the mean percentage of female teachers by all the three methods discussed in this section.
Percentage of female teachersNumber of States/U.T.15-25625-351135-45745-55455-65465-75275-851
(Source: Seventh All India School Education Survey conducted by NCERT)
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Step-by-Step Solution
Step 1: Calculate the Class Mark (xi)
To find the mean using any method, we first need to determine the class mark (xi) for each class interval. The class mark represents the midpoint of each interval and is calculated by averaging the lower and upper limits of the class. This value will be used in our calculations.
Step 2: Direct Method: Calculate Σfixi and Σfi
The direct method for calculating the mean involves summing the products of each frequency (fi) and its corresponding class mark (xi), and then dividing by the sum of all frequencies. This formula provides a straightforward way to find the arithmetic mean of grouped data.
Step 3: Assumed Mean Method: Choose an Assumed Mean (A) and calculate di
In the assumed mean method, we choose an assumed mean (A) from the class marks (usually the middle one) to simplify calculations. We then calculate the deviation (di) of each class mark from this assumed mean. The mean is found by adding the assumed mean to the average of the deviations.
Step 4: Step Deviation Method: Calculate ui
The step deviation method is an extension of the assumed mean method, used when the class intervals have a uniform width (h). We further simplify the deviations by dividing them by the class width to get ui. This makes the numerical values smaller and easier to work with, especially for manual calculations. We then use the given formula to find the mean.
Step 5: Consolidate results in a table for all methods
We consolidate all necessary calculations into a single table. This table includes the class intervals, frequencies (fi), class marks (xi), product fixi, deviations di, step deviations ui, and product fiui. The sums ∑fi, ∑fixi, and ∑fiui are then calculated for use in the mean formulas. We choose A=50 (the class mark of the class with frequency 4) and h=10 (the class width).
Step 6: Calculate Mean by Direct Method
Using the values from our consolidated table, we apply the direct method formula. We substitute the sum of fixi, which is 1390, and the sum of frequencies ∑fi, which is 35. We perform the division to get the mean.
Step 7: Calculate Mean by Assumed Mean Method
For the assumed mean method, we take the assumed mean A=50. We calculate ∑fidi. From the table, ∑fidi=(6×−30)+(11×−20)+(7×−10)+(4×0)+(4×10)+(2×20)+(1×30)=−180−220−70+0+40+40+30=−470+110=−360. We substitute the values into the formula along with ∑fi=35 to find the mean.
Step 8: Calculate Mean by Step Deviation Method
For the step deviation method, we use the assumed mean A=50 and class width h=10. We substitute the value of ∑fiui=−36 and ∑fi=35 into the formula. We perform the calculation to arrive at the mean.
Step 9: Final Calculations
We complete the arithmetic for each method. In the direct method, 1390÷35=39.7142.... For the assumed mean method, we have 50+35−360=50−10.2857...=39.7142.... Similarly, for the step deviation method, 50+(35−36)×10=50−35360=50−10.2857...=39.7142.... All three methods yield the same mean value, approximately 39.71%. We round the result to two decimal places.