The following table shows the ages of the patients admitted in a hospital :
`{:("Age (in years)",5 - 15, 15 - 25, 25 - 35, 35-45, 45-55, 55-65),("No. of case", " "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.

Answer

To solve the problem, we need to find the mode and mean of the given data. Let's break it down step by step.

### Step 1: Organize the Data
We have the following data:

| Age (in years) | No. of cases (f) |
|----------------|-------------------|
| 5 - 15 | 6 |
| 15 - 25 | 11 |
| 25 - 35 | 21 |
| 35 - 45 | 23 |
| 45 - 55 | 14 |
| 55 - 65 | 5 |

### Step 2: Calculate the Mean
To calculate the mean, we will use the formula:

\[ \text{Mean} (\bar{x}) = \frac{\sum (f_i \cdot x_i)}{\sum f_i} \]

Where:
- \( f_i \) = frequency (number of cases)
- \( x_i \) = midpoint of each age group

#### Step 2.1: Find the Midpoint (x_i)
The midpoint for each age group can be calculated as follows:

- For 5 - 15: \( x_1 = \frac{5 + 15}{2} = 10 \)
- For 15 - 25: \( x_2 = \frac{15 + 25}{2} = 20 \)
- For 25 - 35: \( x_3 = \frac{25 + 35}{2} = 30 \)
- For 35 - 45: \( x_4 = \frac{35 + 45}{2} = 40 \)
- For 45 - 55: \( x_5 = \frac{45 + 55}{2} = 50 \)
- For 55 - 65: \( x_6 = \frac{55 + 65}{2} = 60 \)

#### Step 2.2: Create a New Table
Now, we will create a new table to calculate \( f_i \cdot x_i \):

| Age (in years) | No. of cases (f) | Midpoint (x_i) | \( f_i \cdot x_i \) |
|----------------|-------------------|----------------|----------------------|
| 5 - 15 | 6 | 10 | \( 6 \cdot 10 = 60 \) |
| 15 - 25 | 11 | 20 | \( 11 \cdot 20 = 220 \) |
| 25 - 35 | 21 | 30 | \( 21 \cdot 30 = 630 \) |
| 35 - 45 | 23 | 40 | \( 23 \cdot 40 = 920 \) |
| 45 - 55 | 14 | 50 | \( 14 \cdot 50 = 700 \) |
| 55 - 65 | 5 | 60 | \( 5 \cdot 60 = 300 \) |

#### Step 2.3: Calculate the Sums
Now we calculate the sums:

\[ \sum f_i = 6 + 11 + 21 + 23 + 14 + 5 = 80 \]
\[ \sum (f_i \cdot x_i) = 60 + 220 + 630 + 920 + 700 + 300 = 2830 \]

#### Step 2.4: Calculate the Mean
Now we can calculate the mean:

\[ \text{Mean} (\bar{x}) = \frac{2830}{80} = 35.375 \]

### Step 3: Calculate the Mode
To find the mode, we need to identify the class with the highest frequency.

From the table, the highest frequency is 23, which corresponds to the age group 35 - 45.

Using the formula for mode:

\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \cdot h \]

Where:
- \( L = 35 \) (lower limit of modal class)
- \( f_1 = 23 \) (frequency of modal class)
- \( f_0 = 21 \) (frequency of class before modal class)
- \( f_2 = 14 \) (frequency of class after modal class)
- \( h = 10 \) (class width)

Substituting the values:

\[ \text{Mode} = 35 + \left( \frac{23 - 21}{2 \cdot 23 - 21 - 14} \right) \cdot 10 \]
\[ = 35 + \left( \frac{2}{46 - 35} \right) \cdot 10 \]
\[ = 35 + \left( \frac{2}{11} \right) \cdot 10 \]
\[ = 35 + \frac{20}{11} \approx 35 + 1.818 = 36.818 \]

### Step 4: Compare and Interpret
- **Mean**: 35.375
- **Mode**: 36.818

The mean is slightly lower than the mode. This indicates that the data is slightly negatively skewed, meaning there are some lower values that are pulling the mean down compared to the mode, which represents the most frequent age group.