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Magazine Chapter 3: Problem 14
The following data represent the flight time (in minutes) of a random sample of seven flights from Las Vegas Nevada, to Newark, New Jersey, on United Airlines. Compute the range and sample standard deviation of flight time. $$ 282,270,260,266,257,260,267 $$
Short Answer
Expert verified
Range: 25, Sample Standard Deviation: ~8.43
Step by step solution
01
- Arrange the data in ascending order
First, list the flight times in ascending order: 257, 260, 260, 266, 267, 270, 282
02
- Compute the range
The range is the difference between the maximum and minimum values:Range = Maximum value - Minimum valueRange = 282 - 257Range = 25
03
- Compute the mean
To find the sample mean (\(\bar{x}\)), add all the flight times together and then divide by the number of flights:\(\bar{x} = \frac{257 + 260 + 260 + 266 + 267 + 270 + 282}{7} = \frac{1862}{7} = 266.0\)
04
- Calculate each deviation from the mean
Subtract the mean from each flight time and then square the result:(257 - 266.0)^2 = 81.0(260 - 266.0)^2 = 36.0(260 - 266.0)^2 = 36.0(266 - 266.0)^2 = 0.0(267 - 266.0)^2 = 1.0(270 - 266.0)^2 = 16.0(282 - 266.0)^2 = 256.0
05
- Sum the squared deviations
Add all the squared deviations together:81.0 + 36.0 + 36.0 + 0.0 + 1.0 + 16.0 + 256.0 = 426.0
06
- Compute the variance
The sample variance (\(s^2\)) is the sum of the squared deviations divided by the number of observations minus one:\(s^2 = \frac{426.0}{7 - 1} = \frac{426.0}{6} = 71.0\)
07
- Compute the sample standard deviation
The sample standard deviation (\(s\)) is the square root of the variance:\(s = \sqrt{71.0} \approx 8.43\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Range
The range is a simple measure of variability in a data set. It measures the spread between the largest and smallest values. To find the range, subtract the smallest value from the largest value in your data set.
For example, in our flight time data set (257, 260, 260, 266, 267, 270, 282), the range is calculated as follows:
\(\text{Range} = 282 - 257 = 25\).
A larger range indicates more spread in the data, while a smaller range indicates the data points are closer together.
For example, in our flight time data set (257, 260, 260, 266, 267, 270, 282), the range is calculated as follows:
- The minimum value is 257.
- The maximum value is 282.
\(\text{Range} = 282 - 257 = 25\).
A larger range indicates more spread in the data, while a smaller range indicates the data points are closer together.
Sample Standard Deviation
The sample standard deviation is a measure of variability that indicates the average distance between each data point and the mean of the sample.
The steps to calculate the sample standard deviation are:
\(\text{s} = \sqrt{71.0} \: 8.43\).
The sample standard deviation helps understand the spread of the data around the mean.
The steps to calculate the sample standard deviation are:
- Find the mean ( \(\bar{x}\)) of the data set.
- Subtract the mean from each data point and square the result to find the squared deviations.
- Sum all the squared deviations.
- Divide this sum by the number of data points minus one to find the variance.
- Take the square root of the variance to find the sample standard deviation.
\(\text{s} = \sqrt{71.0} \: 8.43\).
The sample standard deviation helps understand the spread of the data around the mean.
Variance
Variance is another measure of spread or dispersion in a data set. It tells how much the data points are spread out from the mean. The variance is the average of the squared differences from the mean.
To calculate the variance ( \(s^2\)), follow these steps:
\(s^2 = \frac{426.0}{6} \: 71.0\).
Variance gives a sense of the variability within the sample data.
To calculate the variance ( \(s^2\)), follow these steps:
- Compute the mean of the data set.
- Find the deviation of each data point from the mean and square these deviations.
- Sum all the squared deviations.
- Divide this sum by the number of observations minus one.
\(s^2 = \frac{426.0}{6} \: 71.0\).
Variance gives a sense of the variability within the sample data.
Mean
The mean, also known as the average, is a measure of central tendency. It represents the typical value in a data set.
To compute the mean ( \(\bar{x}\)), sum all the data points and divide by the number of observations:
\(\bar{x} = \frac{257 + 260 + 260 + 266 + 267 + 270 + 282}{7} \: \frac{1862}{7} = 266.0\).
The mean is useful because it provides a single value that summarizes the data set. It's important to note that the mean can be affected by extreme values (outliers).
To compute the mean ( \(\bar{x}\)), sum all the data points and divide by the number of observations:
\(\bar{x} = \frac{257 + 260 + 260 + 266 + 267 + 270 + 282}{7} \: \frac{1862}{7} = 266.0\).
The mean is useful because it provides a single value that summarizes the data set. It's important to note that the mean can be affected by extreme values (outliers).
Deviation
Deviation measures how far a data point is from the mean. It can tell us about the variation within a data set.
To calculate the deviation for each data point:
\(257 - 266 = -9\).
Squaring these deviations and then averaging them helps calculate the variance. Deviation helps in understanding individual data point disparity from the mean.
To calculate the deviation for each data point:
- Compute the mean of the data set.
- Subtract the mean from each data point.
\(257 - 266 = -9\).
Squaring these deviations and then averaging them helps calculate the variance. Deviation helps in understanding individual data point disparity from the mean.
