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 admitted patients: (using Assumed Mean Method). Mode represents the most frequent age of admitted patients, given by the formula: where is the lower limit of the modal class, is the class size, is the frequency of the modal class, is the frequency of the preceding class, and is the frequency of the succeeding class. Interpretation: Comparing the two measures explains the central tendency of patient ages from two perspectives: the average age versus the age group with the maximum number of admissions. Step 1 · Calculate the Mean Let the assumed mean be . Using the Assumed Mean formula Step 2 · Calculate the Mode The maximum class frequency is , which lies in the class interval . Therefore, the modal class is . Lower limit () Class size () Frequency of modal class () Frequency of preceding class () Frequency of succeeding class () Step 3 · Compare and Interpret the Measures Mode Mean Interpretation: The mode indicates that the maximum number of patients admitted to the hospital are of age years. The mean indicates that the average age of all admitted patients is years. Answer > The maximum number of patients admitted are of age years, while the average age of patients admitted is years. Common Mistakes Index Confusion: Mixing up frequencies: is the modal class, is the preceding class, and is the succeeding class. Denominator Error in Mode: Incorrectly computing the denominator as with wrong signs or omitting the factor of on . Interpretation Confusion: Stating that mean represents the highest frequency instead of mode. Mean is the overall average, whereas mode is the most frequent value.

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 admitted patients: $\bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i}$ (using Assumed Mean Method).
Mode represents the most frequent age of admitted patients, given by the formula: $\text{Mode} = l + \left( \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$ where $l$ is the lower limit of the modal class, $h$ is the class size, $f_1$ is the frequency of the modal class, $f_0$ is the frequency of the preceding class, and $f_2$ is the frequency of the succeeding class.
Interpretation: Comparing the two measures explains the central tendency of patient ages from two perspectives: the average age versus the age group with the maximum number of admissions.

Step 1 · Calculate the Mean

Let the assumed mean be $a = 30$ . 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}

Using the Assumed Mean formula

\begin{aligned} \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 class frequency is $23$ , which lies in the class interval $35 - 45$ .

Therefore, the modal class is $35 - 45$ .

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] &= 35 + 1.81 \\[0.6em] &= 36.81 \text{ years} \end{aligned}

Step 3 · Compare and Interpret the Measures

Mode $= 36.81 \text{ years} \approx 36.8 \text{ years}$
Mean $= 35.38 \text{ years}$

Interpretation:

The mode indicates that the maximum number of patients admitted to the hospital are of age $36.8$ years.
The mean indicates that the average age of all admitted patients is $35.38$ years.

Answer

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

The maximum number of patients admitted are of age $36.8$ years, while the average age of patients admitted is $35.38$ years.

Common Mistakes

Index Confusion: Mixing up frequencies: $f_1$ is the modal class, $f_0$ is the preceding class, and $f_2$ is the succeeding class.
Denominator Error in Mode: Incorrectly computing the denominator as $2f_1 - (f_0 + f_2)$ with wrong signs or omitting the factor of $2$ on $f_1$ .
Interpretation Confusion: Stating that mean represents the highest frequency instead of mode. Mean is the overall average, whereas mode is the most frequent value.

Question 1

More questions in Exercise 13.2