Answer :
To determine the standard deviation and interquartile range (IQR) of the given data set, follow these steps:
### Data Set:
38, 379, 542, 480, 46, 487, 326, 518, 36, 364, 405, 444, 358, 496, 495, 407, 423, 439, 45
### Part (a): Standard Deviation
1. Calculate the Mean:
[tex]\[ \text{Mean} = \frac{\sum x_i}{n} \][/tex]
where [tex]\( x_i \)[/tex] are the individual data points and [tex]\( n \)[/tex] is the number of data points.
- Sum of the data points: [tex]\( 38 + 379 + 542 + 480 + 46 + 487 + 326 + 518 + 36 + 364 + 405 + 444 + 358 + 496 + 495 + 407 + 423 + 439 + 45 \)[/tex]
- Number of data points: [tex]\( n = 19 \)[/tex]
2. Calculate the Variance:
[tex]\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n - 1} \][/tex]
where each [tex]\( x_i \)[/tex] is subtracted from the mean, squared, summed up, and then divided by [tex]\( n - 1 \)[/tex].
3. Standard Deviation:
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} \][/tex]
After going through the calculations, the standard deviation is found to be:
[tex]\[ \text{Standard Deviation} \approx 175.47 \][/tex]
### Part (b): Interquartile Range (IQR)
1. Sort the Data:
Arrange the data in ascending order.
- Sorted Data: [tex]\( 36, 38, 45, 46, 326, 358, 364, 379, 405, 407, 423, 439, 444, 480, 487, 495, 496, 518, 542 \)[/tex]
2. Calculate the Quartiles:
- [tex]\( Q1 \)[/tex] (25th percentile)
- [tex]\( Q3 \)[/tex] (75th percentile)
The quartiles divide the sorted data into four equal parts.
- [tex]\( Q1 \)[/tex] is the median of the first half of the data.
- [tex]\( Q3 \)[/tex] is the median of the second half of the data.
3. Compute IQR:
[tex]\[ \text{IQR} = Q3 - Q1 \][/tex]
After determining the quartiles, the interquartile range (IQR) is found to be:
[tex]\[ \text{IQR} = 141.5 \][/tex]
### Summary:
- Standard Deviation: [tex]\( 175.47 \)[/tex] (rounded to two decimal places)
- Interquartile Range (IQR): [tex]\( 141.5 \)[/tex] (rounded to two decimal places)
These calculations provide a measure of the spread of the data, with the standard deviation indicating the average distance of each data point from the mean, and the IQR indicating the range within which the middle 50% of the data values lie.
### Data Set:
38, 379, 542, 480, 46, 487, 326, 518, 36, 364, 405, 444, 358, 496, 495, 407, 423, 439, 45
### Part (a): Standard Deviation
1. Calculate the Mean:
[tex]\[ \text{Mean} = \frac{\sum x_i}{n} \][/tex]
where [tex]\( x_i \)[/tex] are the individual data points and [tex]\( n \)[/tex] is the number of data points.
- Sum of the data points: [tex]\( 38 + 379 + 542 + 480 + 46 + 487 + 326 + 518 + 36 + 364 + 405 + 444 + 358 + 496 + 495 + 407 + 423 + 439 + 45 \)[/tex]
- Number of data points: [tex]\( n = 19 \)[/tex]
2. Calculate the Variance:
[tex]\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n - 1} \][/tex]
where each [tex]\( x_i \)[/tex] is subtracted from the mean, squared, summed up, and then divided by [tex]\( n - 1 \)[/tex].
3. Standard Deviation:
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} \][/tex]
After going through the calculations, the standard deviation is found to be:
[tex]\[ \text{Standard Deviation} \approx 175.47 \][/tex]
### Part (b): Interquartile Range (IQR)
1. Sort the Data:
Arrange the data in ascending order.
- Sorted Data: [tex]\( 36, 38, 45, 46, 326, 358, 364, 379, 405, 407, 423, 439, 444, 480, 487, 495, 496, 518, 542 \)[/tex]
2. Calculate the Quartiles:
- [tex]\( Q1 \)[/tex] (25th percentile)
- [tex]\( Q3 \)[/tex] (75th percentile)
The quartiles divide the sorted data into four equal parts.
- [tex]\( Q1 \)[/tex] is the median of the first half of the data.
- [tex]\( Q3 \)[/tex] is the median of the second half of the data.
3. Compute IQR:
[tex]\[ \text{IQR} = Q3 - Q1 \][/tex]
After determining the quartiles, the interquartile range (IQR) is found to be:
[tex]\[ \text{IQR} = 141.5 \][/tex]
### Summary:
- Standard Deviation: [tex]\( 175.47 \)[/tex] (rounded to two decimal places)
- Interquartile Range (IQR): [tex]\( 141.5 \)[/tex] (rounded to two decimal places)
These calculations provide a measure of the spread of the data, with the standard deviation indicating the average distance of each data point from the mean, and the IQR indicating the range within which the middle 50% of the data values lie.
