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.

Understand the Question Mean represents the arithmetic average age of all patients admitted to the hospital. Mode represents the most frequent age among the admitted patients (the age group with the maximum number of admissions). We calculate the mean using the assumed mean method: . We calculate the mode using the formula: . Step 1 · Calculate the Mean Using the Assumed Mean method with assumed mean : Now, calculate the mean: Step 2 · Calculate the Mode The maximum frequency is , so the modal class is . From the modal class: Lower limit () Class size () Frequency of modal class () Frequency of preceding class () Frequency of succeeding class () Step 3 · Compare and Interpret Mode : The maximum number of patients admitted to the hospital are aged approximately . Mean : The average age of a patient admitted to the hospital is . Answer > , Common Mistakes Modal Frequency Confusion: Mixing up the indices , , and . Remember: is the modal class frequency, is before it, and is after it. Denominator Sign in Mode Formula: Incorrectly writing instead of . Interpretation Omission: Stating only numerical values without interpreting that mode represents the most frequent age, while mean represents the average age.

Statistics | Exercise 13.2

Question 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.

Solution

Understand the Question

Mean represents the arithmetic average age of all patients admitted to the hospital.
Mode represents the most frequent age among the admitted patients (the age group with the maximum number of admissions).
We calculate the mean using the assumed mean method: $\bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i}$ .
We calculate the mode using the formula: $\text{Mode} = l + \left( \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$ .

Step 1 · Calculate the Mean

Using the Assumed Mean method with assumed mean $a = 30$ : Diagram 1

\begin{array}{|c|c|c|c|c|} \hline \text{Age (in years)} & \text{Number of patients } (f_i) & \text{Class mark } (x_i) & d_i = x_i - 30 & f_i d_i \\ \hline 5 - 15 & 6 & 10 & -20 & -120 \\ \hline 15 - 25 & 11 & 20 & -10 & -110 \\ \hline 25 - 35 & 21 & 30 & 0 & 0 \\ \hline 35 - 45 & 23 & 40 & 10 & 230 \\ \hline 45 - 55 & 14 & 50 & 20 & 280 \\ \hline 55 - 65 & 5 & 60 & 30 & 150 \\ \hline \text{Total} & \sum f_i = 80 & & & \sum f_i d_i = 430 \\ \hline \end{array}

Now, calculate the mean:

\begin{aligned} \text{Mean } (\bar{x}) &= a + \dfrac{\sum f_i d_i}{\sum f_i} \\[0.6em] &= 30 + \dfrac{430}{80} \\[0.6em] &= 30 + 5.375 \\[0.6em] &= 35.38 \text{ years} \end{aligned}

Step 2 · Calculate the Mode

The maximum frequency is $23$ , so the modal class is $35 - 45$ .

From the modal class:

Lower limit ( $l$ ) $= 35$
Class size ( $h$ ) $= 10$
Frequency of modal class ( $f_1$ ) $= 23$
Frequency of preceding class ( $f_0$ ) $= 21$
Frequency of succeeding class ( $f_2$ ) $= 14$

\begin{aligned} \text{Mode} &= l + \left[ \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right] \times h \\[0.6em] &= 35 + \left[ \dfrac{23 - 21}{2(23) - 21 - 14} \right] \times 10 \\[0.6em] &= 35 + \left[ \dfrac{2}{46 - 35} \right] \times 10 \\[0.6em] &= 35 + \dfrac{20}{11} \\[0.6em] &\approx 35 + 1.81 \\[0.6em] &= 36.81 \text{ years (or } 36.8 \text{ years)} \end{aligned}

Step 3 · Compare and Interpret

Mode $= 36.8 \text{ years}$ : The maximum number of patients admitted to the hospital are aged approximately $36.8\text{ years}$ .
Mean $= 35.38 \text{ years}$ : The average age of a patient admitted to the hospital is $35.38\text{ years}$ .

Answer

$\text{Mode} = 36.8\text{ years}$ , $\text{Mean} = 35.38\text{ years}$

Common Mistakes

Modal Frequency Confusion: Mixing up the indices $f_0$ , $f_1$ , and $f_2$ . Remember: $f_1$ is the modal class frequency, $f_0$ is before it, and $f_2$ is after it.
Denominator Sign in Mode Formula: Incorrectly writing $2f_1 + f_0 + f_2$ instead of $2f_1 - f_0 - f_2$ .
Interpretation Omission: Stating only numerical values without interpreting that mode represents the most frequent age, while mean represents the average age.

Question 1

More questions in Exercise 13.2