A Civil Engineer conducted a study on the age of hospital buildings in a city. Calculate the mean, median, and mode.

First, we need to determine the midpoints (x) for each class interval, calculate Fx, and find the cumulative frequencies (Cf). The class width ($w$) is 10 (e.g., $9-0+1=10$). The lower class boundaries ($L$) for calculations are found by subtracting 0.5 from the lower limit of each class (e.g., $20-0.5=19.5$).

Age of building	F (Frequency)	Midpoint (x)	Fx	Cf (Cumulative Frequency)
0-9	19	4.5	85.5	19
10-19	47	14.5	681.5	66
20-29	105	24.5	2572.5	171
30-39	93	34.5	3208.5	264
40-49	16	44.5	712.0	280
Total	N = 280		ΣFx = 7260

1. Calculate the Mean

The mean ($\bar{x}$) is given by:

$$\bar{x} = \frac{\sum Fx}{N}$$

Substitute the total sum of Fx and the total frequency (N) from the table:

$$\bar{x} = \frac{7260}{280}$$ $$\bar{x} \approx 25.93$$

The mean age of the hospital buildings is $\boxed{\text{25.93 years}}$.

2. Calculate the Median

The position of the median is $\frac{N}{2} = \frac{280}{2} = 140$.
The median class is the class where the cumulative frequency (Cf) first exceeds 140. From the table, this is the 20-29 class (Cf = 171).

For the median class (20-29):

• Lower class boundary ($L$) = $20 - 0.5 = 19.5$
• Cumulative frequency of the class before the median class ($Cf_b$) = 66
• Frequency of the median class ($f_m$) = 105
• Class width ($w$) = 10

The formula for the median of grouped data is:

$$Median = L + \left(\frac{\frac{N}{2} - Cf_b}{f_m}\right) \times w$$

Substitute the values:

$$Median = 19.5 + \left(\frac{140 - 66}{105}\right) \times 10$$ $$Median = 19.5 + \left(\frac{74}{105}\right) \times 10$$ $$Median = 19.5 + 0.70476 \times 10$$ $$Median = 19.5 + 7.0476$$ $$Median \approx 26.55$$

The median age of the hospital buildings is $\boxed{\text{26.55 years}}$.

3. Calculate the Mode

The modal class is the class with the highest frequency. From the table, the highest frequency is 105, which corresponds to the 20-29 class.

For the modal class (20-29):

• Lower class boundary ($L$) = $20 - 0.5 = 19.5$
• Frequency of the modal class ($f_m$) = 105
• Frequency of the class before the modal class ($f_1$) = 47 (for 10-19)
• Frequency of the class after the modal class ($f_2$) = 93 (for 30-39)
• Class width ($w$) = 10

The formula for the mode of grouped data is:

$$Mode = L + \left(\frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)}\right) \times w$$

Substitute the values:

$$Mode = 19.5 + \left(\frac{105 - 47}{(105 - 47) + (105 - 93)}\right) \times 10$$ $$Mode = 19.5 + \left(\frac{58}{58 + 12}\right) \times 10$$ $$Mode = 19.5 + \left(\frac{58}{70}\right) \times 10$$ $$Mode = 19.5 + 0.82857 \times 10$$ $$Mode = 19.5 + 8.2857$$ $$Mode \approx 27.79$$

The mode age of the hospital buildings is $\boxed{\text{27.79 years}}$.