The following table shows the ages of the patients admitted in a hospital during a year: Age (in years) 5 - 15 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 Number of patients 6 11 21 23 14 5 Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.
Step-by-Step Solution
Key Concept: For grouped data, the <b>mode</b> is obtained from the modal class using the formula \(\text{Mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h\), where \(L\) is the lower limit of the modal class, \(f_1\) its frequency, \(f_0\) the frequency of the preceding class, \(f_2\) the frequency of the succeeding class and \(h\) the class width. The <b>mean</b> is calculated as \(\bar{x}=\frac{\sum f\,x}{\sum f}\), where \(x\) is the class midpoint.
1. Identify the class intervals and frequencies
\[\begin{array}{c|c|c}
\text{Class (years)} & \text{Midpoint }(x) & \text{Frequency }(f)\\ \hline
5-15 & 10 & 6\\
15-25 & 20 & 11\\
25-35 & 30 & 21\\
35-45 & 40 & 23\\
45-55 & 50 & 14\\
55-65 & 60 & 5\\
\end{array}\]
2. Mode
- The highest frequency is \(f_1=23\) for the class \(35-45\); therefore this is the modal class.
- \(f_0\) (frequency of the preceding class) = 21.
- \(f_2\) (frequency of the succeeding class) = 14.
- Lower limit \(L\) of the modal class = 35.
- Class width \(h\) = 10 (all classes have equal width).
Apply the modal formula:
\[\text{Mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h
=35+\frac{23-21}{2\times23-21-14}\times10
=35+\frac{2}{46-35}\times10
=35+\frac{2}{11}\times10
=35+\frac{20}{11}\approx36.8\text{ years}\]
3. Mean
- Compute \(f\times x\) for each class:
\[\begin{aligned}
6\times10 &= 60\\
11\times20 &= 220\\
21\times30 &= 630\\
23\times40 &= 920\\
14\times50 &= 700\\
5\times60 &= 300\\
\end{aligned}\]
- Sum of \(f\times x\): \(\sum f x = 60+220+630+920+700+300 = 2830\).
- Total number of patients: \(\sum f = 6+11+21+23+14+5 = 80\).
- Mean:
\[\bar{x}=\frac{\sum f x}{\sum f}=\frac{2830}{80}=35.375\approx35.4\text{ years}\]
4. Comparison and Interpretation
- The mode (≈ 36.8 years) indicates that the most frequent age group is the interval 35‑45 years.
- The mean (≈ 35.4 years) gives the average age of all patients.
- Since the mode is slightly greater than the mean, the distribution is slightly negatively (left) skewed – there are a few younger patients pulling the average down, while the largest concentration of patients lies a little higher, around 36‑37 years.
- Both measures are close, showing that the data are fairly symmetric with a mild skewness.
Correct Answer: Mode \(\approx 36.8\) years; Mean \(\approx 35.4\) years. The mode (most frequent age) is a little higher than the mean, indicating a slight left‑skewness of the age distribution.