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Sure! Let's calculate the mean, median, and mode step by step using the given frequency table of heights.
### Step 1: Calculate the Mean
The mean of a set of numbers can be found by dividing the sum of all data points by the total number of data points.
1. Sum of all observed data points:
We need to multiply each height by its corresponding frequency and then sum up the results.
[tex]\[ \sum (\text{Height} \times \text{Frequency}) = 63 \times 2 + 65 \times 1 + 66 \times 4 + 67 \times 3 + 68 \times 1 + 69 \times 2 + 70 \times 2 + 71 \times 1 + 72 \times 3 + 74 \times 2 + 75 \times 2 \][/tex]
[tex]\[ = 126 + 65 + 264 + 201 + 68 + 138 + 140 + 71 + 216 + 148 + 150 = 1587 \][/tex]
2. Total number of observations:
[tex]\[ n = \sum \text{Frequency} = 2 + 1 + 4 + 3 + 1 + 2 + 2 + 1 + 3 + 2 + 2 = 23 \][/tex]
3. Mean Calculation:
[tex]\[ \text{Mean} = \frac{\sum (\text{Height} \times \text{Frequency})}{n} = \frac{1587}{23} \approx 69 \][/tex]
### Step 2: Determine the Median
The median is the value that separates the data into two equal halves. In a frequency distribution, it is useful to construct a cumulative frequency to locate the median position.
1. Cumulative Frequency:
[tex]\[ \begin{array}{c|c|c} \text{Inches} & \text{Frequency} & \text{Cumulative Frequency} \\ \hline 63 & 2 & 2 \\ 65 & 1 & 3 \\ 66 & 4 & 7 \\ 67 & 3 & 10 \\ 68 & 1 & 11 \\ 69 & 2 & 13 \\ 70 & 2 & 15 \\ 71 & 1 & 16 \\ 72 & 3 & 19 \\ 74 & 2 & 21 \\ 75 & 2 & 23 \\ \end{array} \][/tex]
2. Locate the median position:
Since there are 23 observations (an odd number), the median will be the value at the 12th position.
From the cumulative frequency chart, we see that the 12th position lies within the height category of 69 inches.
[tex]\[ \text{Median} = 69 \text{ inches} \][/tex]
### Step 3: Determine the Mode
The mode is the value(s) that appear most frequently in the data set.
1. Identify the maximum frequency:
In this table, the highest frequency is 4, which corresponds to the height of 66 inches.
[tex]\[ \text{Mode} = 66 \text{ inches} \][/tex]
### Summary of Results
- Mean: [tex]\( 69 \)[/tex] inches
- Median: [tex]\( 69 \)[/tex] inches
- Mode: [tex]\( 66 \)[/tex] inches
By following these calculations precisely, we have determined the mean, median, and mode of the given frequency distribution of adult male heights.
### Step 1: Calculate the Mean
The mean of a set of numbers can be found by dividing the sum of all data points by the total number of data points.
1. Sum of all observed data points:
We need to multiply each height by its corresponding frequency and then sum up the results.
[tex]\[ \sum (\text{Height} \times \text{Frequency}) = 63 \times 2 + 65 \times 1 + 66 \times 4 + 67 \times 3 + 68 \times 1 + 69 \times 2 + 70 \times 2 + 71 \times 1 + 72 \times 3 + 74 \times 2 + 75 \times 2 \][/tex]
[tex]\[ = 126 + 65 + 264 + 201 + 68 + 138 + 140 + 71 + 216 + 148 + 150 = 1587 \][/tex]
2. Total number of observations:
[tex]\[ n = \sum \text{Frequency} = 2 + 1 + 4 + 3 + 1 + 2 + 2 + 1 + 3 + 2 + 2 = 23 \][/tex]
3. Mean Calculation:
[tex]\[ \text{Mean} = \frac{\sum (\text{Height} \times \text{Frequency})}{n} = \frac{1587}{23} \approx 69 \][/tex]
### Step 2: Determine the Median
The median is the value that separates the data into two equal halves. In a frequency distribution, it is useful to construct a cumulative frequency to locate the median position.
1. Cumulative Frequency:
[tex]\[ \begin{array}{c|c|c} \text{Inches} & \text{Frequency} & \text{Cumulative Frequency} \\ \hline 63 & 2 & 2 \\ 65 & 1 & 3 \\ 66 & 4 & 7 \\ 67 & 3 & 10 \\ 68 & 1 & 11 \\ 69 & 2 & 13 \\ 70 & 2 & 15 \\ 71 & 1 & 16 \\ 72 & 3 & 19 \\ 74 & 2 & 21 \\ 75 & 2 & 23 \\ \end{array} \][/tex]
2. Locate the median position:
Since there are 23 observations (an odd number), the median will be the value at the 12th position.
From the cumulative frequency chart, we see that the 12th position lies within the height category of 69 inches.
[tex]\[ \text{Median} = 69 \text{ inches} \][/tex]
### Step 3: Determine the Mode
The mode is the value(s) that appear most frequently in the data set.
1. Identify the maximum frequency:
In this table, the highest frequency is 4, which corresponds to the height of 66 inches.
[tex]\[ \text{Mode} = 66 \text{ inches} \][/tex]
### Summary of Results
- Mean: [tex]\( 69 \)[/tex] inches
- Median: [tex]\( 69 \)[/tex] inches
- Mode: [tex]\( 66 \)[/tex] inches
By following these calculations precisely, we have determined the mean, median, and mode of the given frequency distribution of adult male heights.
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