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Sagot :
Sure, I'll guide you through the process of constructing a discrete frequency table and finding the mean deviation from the median for the given set of data.
### Step 1: Organize the Data into a Frequency Table
First, let's list down the data values and their corresponding frequencies.
Given data:
[tex]$19, 23, 30, 29, 11, 21, 26, 36, 41, 42, 49, 52, 56, 58, 53, 27, 20, 34, 62, 22, 23, 25, 27, 36, 42, 42, 52, 50, 58, 53, 30, 11, 29, 26, 41, 36, 58, 53, 19, 23, 30$[/tex]
### Frequency Table
List each data value and count its frequency. This is the frequency distribution:
| Value | Frequency |
|-------|-----------|
| 11 | 2 |
| 19 | 2 |
| 20 | 1 |
| 21 | 1 |
| 22 | 1 |
| 23 | 3 |
| 25 | 1 |
| 26 | 2 |
| 27 | 2 |
| 29 | 2 |
| 30 | 3 |
| 34 | 1 |
| 36 | 3 |
| 41 | 2 |
| 42 | 3 |
| 49 | 1 |
| 50 | 1 |
| 52 | 2 |
| 53 | 3 |
| 56 | 1 |
| 58 | 3 |
| 62 | 1 |
### Step 2: Calculate the Median
Sort the data values in ascending order:
[tex]$11, 11, 19, 19, 20, 21, 22, 23, 23, 23, 25, 26, 26, 27, 27, 29, 29, 30, 30, 30, 34, 36, 36, 36, 41, 41, 42, 42, 42, 49, 50, 52, 52, 53, 53, 53, 56, 58, 58, 58, 62$[/tex]
The number of data points [tex]\(n = 41\)[/tex] (an odd number).
Median is the middle value in an ordered list. For [tex]\(n = 41\)[/tex], the median is the [tex]\( \left(\frac{41 + 1}{2}\right) = 21^{st} \)[/tex] value:
- 21st value: [tex]\( 34 \)[/tex]
So, the median is [tex]\(34\)[/tex].
### Step 3: Calculate the Mean Deviation from Median
Mean deviation from median is given by:
[tex]\[ \text{Mean Deviation} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \text{Median}| \][/tex]
Summing the absolute deviations from the median [tex]\(34\)[/tex]:
[tex]\[ |11 - 34| = 23 \][/tex]
[tex]\[ |11 - 34| = 23 \][/tex]
[tex]\[ |19 - 34| = 15 \][/tex]
[tex]\[ |19 - 34| = 15 \][/tex]
[tex]\[ |20 - 34| = 14 \][/tex]
[tex]\[ |21 - 34| = 13 \][/tex]
[tex]\[ |22 - 34| = 12 \][/tex]
[tex]\[ |23 - 34| = 11 \][/tex]
[tex]\[ |23 - 34| = 11 \][/tex]
[tex]\[ |23 - 34| = 11 \][/tex]
[tex]\[ |25 - 34| = 9 \][/tex]
[tex]\[ |26 - 34| = 8 \][/tex]
[tex]\[ |26 - 34| = 8 \][/tex]
[tex]\[ |27 - 34| = 7 \][/tex]
[tex]\[ |27 - 34| = 7 \][/tex]
[tex]\[ |29 - 34| = 5 \][/tex]
[tex]\[ |29 - 34| = 5 \][/tex]
[tex]\[ |30 - 34| = 4 \][/tex]
[tex]\[ |30 - 34| = 4 \][/tex]
[tex]\[ |30 - 34| = 4 \][/tex]
[tex]\[ |34 - 34| = 0 \][/tex]
[tex]\[ |36 - 34| = 2 \][/tex]
[tex]\[ |36 - 34| = 2 \][/tex]
[tex]\[ |36 - 34| = 2 \][/tex]
[tex]\[ |41 - 34| = 7 \][/tex]
[tex]\[ |41 - 34| = 7 \][/tex]
[tex]\[ |42 - 34| = 8 \][/tex]
[tex]\[ |42 - 34| = 8 \][/tex]
[tex]\[ |42 - 34| = 8 \][/tex]
[tex]\[ |49 - 34| = 15 \][/tex]
[tex]\[ |50 - 34| = 16 \][/tex]
[tex]\[ |52 - 34| = 18 \][/tex]
[tex]\[ |52 - 34| = 18 \][/tex]
[tex]\[ |53 - 34| = 19 \][/tex]
[tex]\[ |53 - 34| = 19 \][/tex]
[tex]\[ |53 - 34| = 19 \][/tex]
[tex]\[ |56 - 34| = 22 \][/tex]
[tex]\[ |58 - 34| = 24 \][/tex]
[tex]\[ |58 - 34| = 24 \][/tex]
[tex]\[ |58 - 34| = 24 \][/tex]
[tex]\[ |62 - 34| = 28 \][/tex]
Summing these values, we get the total absolute deviation:
[tex]\[ \sum |x_i - 34| = 499 \][/tex]
Finally, the mean deviation from the median is:
[tex]\[ \frac{499}{41} \approx 12.170731707317072 \][/tex]
### Answer Summary
- Frequency distribution: \[tex]\({19: 2, 23: 3, 30: 3, 29: 2, 11: 2, 21: 1, 26: 2, 36: 3, 41: 2, 42: 3, 49: 1, 52: 2, 56: 1, 58: 3, 53: 3, 27: 2, 20: 1, 34: 1, 62: 1, 22: 1, 25: 1, 50: 1}\\)[/tex]
- Median: [tex]\(34\)[/tex]
- Mean deviation from the median: [tex]\(12.170731707317072\)[/tex]
### Step 1: Organize the Data into a Frequency Table
First, let's list down the data values and their corresponding frequencies.
Given data:
[tex]$19, 23, 30, 29, 11, 21, 26, 36, 41, 42, 49, 52, 56, 58, 53, 27, 20, 34, 62, 22, 23, 25, 27, 36, 42, 42, 52, 50, 58, 53, 30, 11, 29, 26, 41, 36, 58, 53, 19, 23, 30$[/tex]
### Frequency Table
List each data value and count its frequency. This is the frequency distribution:
| Value | Frequency |
|-------|-----------|
| 11 | 2 |
| 19 | 2 |
| 20 | 1 |
| 21 | 1 |
| 22 | 1 |
| 23 | 3 |
| 25 | 1 |
| 26 | 2 |
| 27 | 2 |
| 29 | 2 |
| 30 | 3 |
| 34 | 1 |
| 36 | 3 |
| 41 | 2 |
| 42 | 3 |
| 49 | 1 |
| 50 | 1 |
| 52 | 2 |
| 53 | 3 |
| 56 | 1 |
| 58 | 3 |
| 62 | 1 |
### Step 2: Calculate the Median
Sort the data values in ascending order:
[tex]$11, 11, 19, 19, 20, 21, 22, 23, 23, 23, 25, 26, 26, 27, 27, 29, 29, 30, 30, 30, 34, 36, 36, 36, 41, 41, 42, 42, 42, 49, 50, 52, 52, 53, 53, 53, 56, 58, 58, 58, 62$[/tex]
The number of data points [tex]\(n = 41\)[/tex] (an odd number).
Median is the middle value in an ordered list. For [tex]\(n = 41\)[/tex], the median is the [tex]\( \left(\frac{41 + 1}{2}\right) = 21^{st} \)[/tex] value:
- 21st value: [tex]\( 34 \)[/tex]
So, the median is [tex]\(34\)[/tex].
### Step 3: Calculate the Mean Deviation from Median
Mean deviation from median is given by:
[tex]\[ \text{Mean Deviation} = \frac{1}{n} \sum_{i=1}^{n} |x_i - \text{Median}| \][/tex]
Summing the absolute deviations from the median [tex]\(34\)[/tex]:
[tex]\[ |11 - 34| = 23 \][/tex]
[tex]\[ |11 - 34| = 23 \][/tex]
[tex]\[ |19 - 34| = 15 \][/tex]
[tex]\[ |19 - 34| = 15 \][/tex]
[tex]\[ |20 - 34| = 14 \][/tex]
[tex]\[ |21 - 34| = 13 \][/tex]
[tex]\[ |22 - 34| = 12 \][/tex]
[tex]\[ |23 - 34| = 11 \][/tex]
[tex]\[ |23 - 34| = 11 \][/tex]
[tex]\[ |23 - 34| = 11 \][/tex]
[tex]\[ |25 - 34| = 9 \][/tex]
[tex]\[ |26 - 34| = 8 \][/tex]
[tex]\[ |26 - 34| = 8 \][/tex]
[tex]\[ |27 - 34| = 7 \][/tex]
[tex]\[ |27 - 34| = 7 \][/tex]
[tex]\[ |29 - 34| = 5 \][/tex]
[tex]\[ |29 - 34| = 5 \][/tex]
[tex]\[ |30 - 34| = 4 \][/tex]
[tex]\[ |30 - 34| = 4 \][/tex]
[tex]\[ |30 - 34| = 4 \][/tex]
[tex]\[ |34 - 34| = 0 \][/tex]
[tex]\[ |36 - 34| = 2 \][/tex]
[tex]\[ |36 - 34| = 2 \][/tex]
[tex]\[ |36 - 34| = 2 \][/tex]
[tex]\[ |41 - 34| = 7 \][/tex]
[tex]\[ |41 - 34| = 7 \][/tex]
[tex]\[ |42 - 34| = 8 \][/tex]
[tex]\[ |42 - 34| = 8 \][/tex]
[tex]\[ |42 - 34| = 8 \][/tex]
[tex]\[ |49 - 34| = 15 \][/tex]
[tex]\[ |50 - 34| = 16 \][/tex]
[tex]\[ |52 - 34| = 18 \][/tex]
[tex]\[ |52 - 34| = 18 \][/tex]
[tex]\[ |53 - 34| = 19 \][/tex]
[tex]\[ |53 - 34| = 19 \][/tex]
[tex]\[ |53 - 34| = 19 \][/tex]
[tex]\[ |56 - 34| = 22 \][/tex]
[tex]\[ |58 - 34| = 24 \][/tex]
[tex]\[ |58 - 34| = 24 \][/tex]
[tex]\[ |58 - 34| = 24 \][/tex]
[tex]\[ |62 - 34| = 28 \][/tex]
Summing these values, we get the total absolute deviation:
[tex]\[ \sum |x_i - 34| = 499 \][/tex]
Finally, the mean deviation from the median is:
[tex]\[ \frac{499}{41} \approx 12.170731707317072 \][/tex]
### Answer Summary
- Frequency distribution: \[tex]\({19: 2, 23: 3, 30: 3, 29: 2, 11: 2, 21: 1, 26: 2, 36: 3, 41: 2, 42: 3, 49: 1, 52: 2, 56: 1, 58: 3, 53: 3, 27: 2, 20: 1, 34: 1, 62: 1, 22: 1, 25: 1, 50: 1}\\)[/tex]
- Median: [tex]\(34\)[/tex]
- Mean deviation from the median: [tex]\(12.170731707317072\)[/tex]
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