Statistics | Exercise 13.2

Question 4

The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.

Question diagram 1
Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • Mode represents the most frequent teacher-student ratio among the states/U.T.
  • Mean represents the overall average teacher-student ratio across all states/U.T.
  • To find the Mode, identify the modal class (the class with the highest frequency) and apply the formula: Mode=l+(f1−f02f1−f0−f2)×h\text{Mode} = l + \left( \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h
  • To find the Mean, use the Step-Deviation method: xˉ=a+(∑fiui∑fi)×h\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h

Step 1 · Calculate the Mode

The maximum frequency is 1010, which corresponds to the class interval 30−3530 - 35.

Therefore, the modal class is 30−3530 - 35.

  • Lower limit of modal class (ll) =30= 30
  • Class size (hh) =5= 5
  • Frequency of modal class (f1f_1) =10= 10
  • Frequency of preceding class (f0f_0) =9= 9
  • Frequency of succeeding class (f2f_2) =3= 3Diagram 1

Using the mode formula: Mode=l+[f1−f02f1−f0−f2]×h\text{Mode} = l + \left[ \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right] \times h

Mode=30+[10−92(10)−9−3]×5=30+[120−12]×5=30+[18]×5=30+0.125×5=30+0.625=30.625\begin{aligned} \text{Mode} &= 30 + \left[ \dfrac{10 - 9}{2(10) - 9 - 3} \right] \times 5 \\[0.6em] &= 30 + \left[ \dfrac{1}{20 - 12} \right] \times 5 \\[0.6em] &= 30 + \left[ \dfrac{1}{8} \right] \times 5 \\[0.6em] &= 30 + 0.125 \times 5 \\[0.6em] &= 30 + 0.625 \\[0.6em] &= 30.625 \end{aligned}

Step 2 · Calculate the Mean using Step-Deviation Method

Let Assumed Mean a=32.5a = 32.5 and class size h=5h = 5.

Number of students per teacherNumber of states/U.T. (fi)xidi=xi−32.5ui=di5fiui15−20317.5−15−3−920−25822.5−10−2−1625−30927.5−5−1−930−351032.500035−40337.551340−45042.5102045−50047.5153050−55252.52048Total35−23\begin{array}{|c|c|c|c|c|c|} \hline \text{Number of students per teacher} & \text{Number of states/U.T. } (f_i) & x_i & d_i = x_i - 32.5 & u_i = \dfrac{d_i}{5} & f_i u_i \\ \hline 15 - 20 & 3 & 17.5 & -15 & -3 & -9 \\ \hline 20 - 25 & 8 & 22.5 & -10 & -2 & -16 \\ \hline 25 - 30 & 9 & 27.5 & -5 & -1 & -9 \\ \hline 30 - 35 & 10 & 32.5 & 0 & 0 & 0 \\ \hline 35 - 40 & 3 & 37.5 & 5 & 1 & 3 \\ \hline 40 - 45 & 0 & 42.5 & 10 & 2 & 0 \\ \hline 45 - 50 & 0 & 47.5 & 15 & 3 & 0 \\ \hline 50 - 55 & 2 & 52.5 & 20 & 4 & 8 \\ \hline \mathbf{\text{Total}} & \mathbf{35} & & & & \mathbf{-23} \\ \hline \end{array}

From the table: ∑fi=35,∑fiui=−23\sum f_i = 35, \quad \sum f_i u_i = -23

Using the Step-Deviation formula: xˉ=a+(∑fiui∑fi)×h\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h

xˉ=32.5+(−2335)×5=32.5−237=32.5−3.2857…≈29.22\begin{aligned} \bar{x} &= 32.5 + \left( \dfrac{-23}{35} \right) \times 5 \\[0.6em] &= 32.5 - \dfrac{23}{7} \\[0.6em] &= 32.5 - 3.2857\dots \\[0.6em] &\approx 29.22 \end{aligned}
Answer
  • Mode =30.625= 30.625 (or ≈30.6\approx 30.6)
  • Mean ≈29.22\approx 29.22 (or ≈29.2\approx 29.2)
  • Interpretation: Most states/U.T. have a teacher-student ratio of approximately 30.630.6, while the average teacher-student ratio across all states/U.T. is approximately 29.229.2.
Common Mistakes
  • Swapping Frequencies: Confusing f0f_0 (preceding frequency) and f2f_2 (succeeding frequency) in the mode formula.
  • Denominator Sign in Mode: Incorrectly writing the denominator as 2f1+f0+f22f_1 + f_0 + f_2 instead of 2f1−f0−f22f_1 - f_0 - f_2.
  • Sign Errors in ∑fiui\sum f_i u_i: Neglecting negative signs when summing negative and positive fiuif_i u_i values.
  • Forgetting hh: Omitting the class size h=5h = 5 while applying the step-deviation or mode formula.

More questions in Exercise 13.2

Q1
  1. The following table shows the ages of the patients admitted in a hospital during a year:

Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Q2
  1. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components :

Determine the modal lifetimes of the components.

Q3
  1. The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure :
Q4

The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.

Q5

The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.

Find the mode of the data.

Q6

A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data :

← Back to Statistics