A researcher given two tests- one in English language and another in Mathematics. He/She finds the mean and standard deviation of each of the two groups n = 100, as follows:

	English	Mathematics
Mean	50	60
Standard deviation	10	12

 

Assuming the curve for the two distributions overlapping, how many students in English language are below the mean of those in Mathematics?

The curve for the two distributions is

We know that One standard deviation, or one sigma, plotted above or below the average value on that normal distribution curve, would define a region that includes 68 percent of all the data points. Two sigmas above or below would include about 95 percent of the data, and three sigmas would include 99.7 percent. This can be understood by the graph given below:

Therefore, the value between the means of English and Mathematics is µ + 1\(\sigma\) = 34.1 %

The number of students in the shaded region given in the above graph = \({Percentage\ of\ the\ shaded\ region \over 100} \times total\ number\ of\ students\ (n)\)

⇒ the number of students in the shaded region = \({34.1 \over100} \times 100\)

⇒ the number of students in the shaded region = 34.1 ≈ 34 (No. of students is discrete value)

Important Points

We need to find the number of students in the English language below the mean of those in Mathematics i.e the portion below the mean of mathematics.

⇒ 50% + 34 = 50 + 34 = 84

Hence 84 students in English language are below the mean of those in Mathematics.