Answer :
Certainly! Let's determine the mean, median, and mode of the given data for adult male heights. We will break down each step carefully.
### Data:
[tex]\[ \begin{array}{l|c} \text{Inches} & \text{Frequency} \\ \hline 63 & 2 \\ 65 & 1 \\ 66 & 4 \\ 67 & 3 \\ 68 & 1 \\ 69 & 2 \\ 70 & 2 \\ 71 & 1 \\ 72 & 3 \\ 74 & 2 \\ 75 & 2 \\ \end{array} \][/tex]
### Step 1: Calculate the Mean
The mean (average) is calculated by dividing the sum of all heights (each height multiplied by its frequency) by the total number of observations (frequencies).
1. Calculate the sum of heights multiplied by their frequencies:
[tex]\[ (63 \times 2) + (65 \times 1) + (66 \times 4) + (67 \times 3) + (68 \times 1) + (69 \times 2) + (70 \times 2) + (71 \times 1) + (72 \times 3) + (74 \times 2) + (75 \times 2) \][/tex]
2. Perform the multiplications:
[tex]\[ 126 + 65 + 264 + 201 + 68 + 138 + 140 + 71 + 216 + 148 + 150 = 1587 \][/tex]
3. Calculate the total number of observations (frequencies):
[tex]\[ 2 + 1 + 4 + 3 + 1 + 2 + 2 + 1 + 3 + 2 + 2 = 23 \][/tex]
4. Divide the total sum by the number of observations:
[tex]\[ \frac{1587}{23} = 69.0 \][/tex]
So, the mean height is [tex]\( 69.0 \)[/tex] inches.
### Step 2: Calculate the Median
The median is the middle value in the list of numbers. If the list has an odd number of observations, the median is the center number. If even, it's the average of the two center numbers.
1. First, list all heights according to their frequencies in ascending order:
[tex]\[ [63, 63, 65, 66, 66, 66, 66, 67, 67, 67, 68, 69, 69, 70, 70, 71, 72, 72, 72, 74, 74, 75, 75] \][/tex]
2. Count the total number of heights (already determined as 23, which is odd).
3. Find the middle position which is [tex]\( \frac{23 + 1}{2} \)[/tex]:
[tex]\[ \text{Middle position} = 12 \][/tex]
4. The height at the 12th position in the ordered list is:
[tex]\[ 69 \][/tex]
So, the median height is [tex]\( 69 \)[/tex] inches.
### Step 3: Calculate the Mode
The mode is the height that appears most frequently in the data.
1. Examine the frequencies:
[tex]\[ \begin{array}{l|c} \text{Inches} & \text{Frequency} \\ \hline 63 & 2 \\ 65 & 1 \\ 66 & 4 \\ 67 & 3 \\ 68 & 1 \\ 69 & 2 \\ 70 & 2 \\ 71 & 1 \\ 72 & 3 \\ 74 & 2 \\ 75 & 2 \end{array} \][/tex]
2. Identify the height with the highest frequency:
[tex]\[ \text{Height 66 has the highest frequency of 4} \][/tex]
So, the mode height is [tex]\( 66 \)[/tex] inches.
### Summary
- Mean: [tex]\( 69.0 \)[/tex] inches
- Median: [tex]\( 69 \)[/tex] inches
- Mode: [tex]\( 66 \)[/tex] inches
### Data:
[tex]\[ \begin{array}{l|c} \text{Inches} & \text{Frequency} \\ \hline 63 & 2 \\ 65 & 1 \\ 66 & 4 \\ 67 & 3 \\ 68 & 1 \\ 69 & 2 \\ 70 & 2 \\ 71 & 1 \\ 72 & 3 \\ 74 & 2 \\ 75 & 2 \\ \end{array} \][/tex]
### Step 1: Calculate the Mean
The mean (average) is calculated by dividing the sum of all heights (each height multiplied by its frequency) by the total number of observations (frequencies).
1. Calculate the sum of heights multiplied by their frequencies:
[tex]\[ (63 \times 2) + (65 \times 1) + (66 \times 4) + (67 \times 3) + (68 \times 1) + (69 \times 2) + (70 \times 2) + (71 \times 1) + (72 \times 3) + (74 \times 2) + (75 \times 2) \][/tex]
2. Perform the multiplications:
[tex]\[ 126 + 65 + 264 + 201 + 68 + 138 + 140 + 71 + 216 + 148 + 150 = 1587 \][/tex]
3. Calculate the total number of observations (frequencies):
[tex]\[ 2 + 1 + 4 + 3 + 1 + 2 + 2 + 1 + 3 + 2 + 2 = 23 \][/tex]
4. Divide the total sum by the number of observations:
[tex]\[ \frac{1587}{23} = 69.0 \][/tex]
So, the mean height is [tex]\( 69.0 \)[/tex] inches.
### Step 2: Calculate the Median
The median is the middle value in the list of numbers. If the list has an odd number of observations, the median is the center number. If even, it's the average of the two center numbers.
1. First, list all heights according to their frequencies in ascending order:
[tex]\[ [63, 63, 65, 66, 66, 66, 66, 67, 67, 67, 68, 69, 69, 70, 70, 71, 72, 72, 72, 74, 74, 75, 75] \][/tex]
2. Count the total number of heights (already determined as 23, which is odd).
3. Find the middle position which is [tex]\( \frac{23 + 1}{2} \)[/tex]:
[tex]\[ \text{Middle position} = 12 \][/tex]
4. The height at the 12th position in the ordered list is:
[tex]\[ 69 \][/tex]
So, the median height is [tex]\( 69 \)[/tex] inches.
### Step 3: Calculate the Mode
The mode is the height that appears most frequently in the data.
1. Examine the frequencies:
[tex]\[ \begin{array}{l|c} \text{Inches} & \text{Frequency} \\ \hline 63 & 2 \\ 65 & 1 \\ 66 & 4 \\ 67 & 3 \\ 68 & 1 \\ 69 & 2 \\ 70 & 2 \\ 71 & 1 \\ 72 & 3 \\ 74 & 2 \\ 75 & 2 \end{array} \][/tex]
2. Identify the height with the highest frequency:
[tex]\[ \text{Height 66 has the highest frequency of 4} \][/tex]
So, the mode height is [tex]\( 66 \)[/tex] inches.
### Summary
- Mean: [tex]\( 69.0 \)[/tex] inches
- Median: [tex]\( 69 \)[/tex] inches
- Mode: [tex]\( 66 \)[/tex] inches
