Let's solve the question step by step. We are given a frequency distribution of the ages of patients admitted to a hospital. Here's the data:
Age Group (Years): 0-15, 15-25, 25-35, 35-45, 45-55, 55-65
Number of Patients: 6, 11, 12, 13, 14, 5
Step 1: Finding the Mode
The mode is the value that appears most frequently in a data set. In a grouped frequency distribution, the mode can be calculated using the formula:
Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] * h
Where:
L = lower boundary of the modal class
f₁ = frequency of the modal class
f₀ = frequency of the class before the modal class
f₂ = frequency of the class after the modal class
h = class width
From the data, we observe that the highest frequency occurs in the class 35-45, where the number of patients is 14. This is the modal class.
Now, we calculate the mode:
L = 35 (lower boundary of the modal class)
f₁ = 14 (frequency of the modal class)
f₀ = 13 (frequency of the class before the modal class, i.e., 25-35)
f₂ = 5 (frequency of the class after the modal class, i.e., 45-55)
h = 10 (class width)
Substituting these values into the formula:
Mode = 35 + [(14 - 13) / (2*14 - 13 - 5)] * 10 Mode = 35 + [1 / (28 - 18)] * 10 Mode = 35 + [1 / 10] * 10 Mode = 35 + 1 Mode = 36
Thus, the mode is 36 years.
Step 2: Finding the Mean
The mean is the average of the data. To find the mean of a grouped frequency distribution, we use the following formula:
Mean = (Σ(f * x)) / Σf
Where:
f = frequency of the class
x = midpoint of the class
Σ(f * x) is the sum of the products of frequencies and midpoints
Σf is the total number of patients
To calculate the midpoints (x) for each class:
For the class 0-15, midpoint x = (0 + 15) / 2 = 7.5
For the class 15-25, midpoint x = (15 + 25) / 2 = 20
For the class 25-35, midpoint x = (25 + 35) / 2 = 30
For the class 35-45, midpoint x = (35 + 45) / 2 = 40
For the class 45-55, midpoint x = (45 + 55) / 2 = 50
For the class 55-65, midpoint x = (55 + 65) / 2 = 60
Now, we calculate the sum of (f * x) for each class:
6 * 7.5 = 45
11 * 20 = 220
12 * 30 = 360
13 * 40 = 520
14 * 50 = 700
5 * 60 = 300
Σ(f * x) = 45 + 220 + 360 + 520 + 700 + 300 = 2145
Next, calculate Σf (the total number of patients):
Σf = 6 + 11 + 12 + 13 + 14 + 5 = 61
Now, calculate the mean:
Mean = Σ(f * x) / Σf Mean = 2145 / 61 Mean ≈ 35.2
Thus, the mean is approximately 35.2 years.
Step 3: Comparing and Interpreting the Mode and Mean
The mode of the data is 36 years, which represents the most frequent age group of patients.
The mean of the data is 35.2 years, which is the average age of all patients.
Comparison:
The mode (36) is slightly higher than the mean (35.2). This indicates that the distribution of ages is slightly skewed to the left (negative skew), meaning there are more younger patients compared to older ones.
Both measures indicate that the most common age range is around 35 years, but the mean provides the overall average, while the mode gives the age group that occurs most frequently.
In summary, the mean and mode are close, and the distribution appears fairly symmetrical, with a slight skew towards younger patients.
