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Chapter 3: Problem 15

Consider a sample with data values of \(27,25,20,15,30,34,28,\) and \(25 .\) Compute the range, interquartile range, variance, and standard deviation.

Short Answer

Expert verified
Range = 19, IQR = 6.5, Variance = 30.25, Standard Deviation 鈮� 5.5.

Step by step solution

01

Arrange the Data in Ascending Order

First, arrange the data values in ascending order: 15, 20, 25, 25, 27, 28, 30, 34.
02

Calculate the Range

The range is the difference between the maximum and minimum values in the dataset. Subtract the smallest value from the largest value: Range = 34 - 15 = 19.
03

Find the Median and Quartiles

To find the interquartile range, first determine the median and quartiles. Since there are 8 data points, the median is the average of the 4th and 5th values: Median = \(\frac{(25 + 27)}{2} = 26\).The first quartile (\(Q_1\)) is the median of the first half: \(Q_1 = \frac{20 + 25}{2} = 22.5\).The third quartile (\(Q_3\)) is the median of the second half: \(Q_3 = \frac{28 + 30}{2} = 29\).
04

Calculate the Interquartile Range (IQR)

The interquartile range (IQR) is the difference between the third quartile \(Q_3\) and the first quartile \(Q_1\).IQR = 29 - 22.5 = 6.5.
05

Calculate the Mean

The mean of the dataset is calculated by summing all data points and dividing by their number.Mean = \(\frac{15 + 20 + 25 + 25 + 27 + 28 + 30 + 34}{8} = \frac{204}{8} = 25.5\).
06

Calculate the Variance

Variance measures the dispersion of the dataset. First, find the squared differences between each data point and the mean:\((x_i - 25.5)^2\).- (15 - 25.5)^2 = 110.25- (20 - 25.5)^2 = 30.25- (25 - 25.5)^2 = 0.25- (25 - 25.5)^2 = 0.25- (27 - 25.5)^2 = 2.25- (28 - 25.5)^2 = 6.25- (30 - 25.5)^2 = 20.25- (34 - 25.5)^2 = 72.25Next, divide the sum of these squared differences by the total number of data points:Variance = \(\frac{110.25 + 30.25 + 0.25 + 0.25 + 2.25 + 6.25 + 20.25 + 72.25}{8} = \frac{242}{8} = 30.25\).
07

Calculate the Standard Deviation

The standard deviation is the square root of the variance. Standard Deviation = \(\sqrt{30.25} \approx 5.5\).

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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 how spread out values in a dataset are. It tells us the difference between the smallest and the largest data values.
To find the range, you need to subtract the smallest number in the dataset from the largest number. It's straightforward but gives a quick sense of the dataset's variability.
  • For the given dataset (15, 20, 25, 25, 27, 28, 30, 34), the smallest value is 15, and the largest is 34.
  • Thus, the range is calculated as: Range = 34 - 15 = 19.
While informative, the range has its limitations as it only considers the extreme values and might not represent the overall dataset's variability.
Interquartile Range
The interquartile range (IQR) measures the spread of the middle 50% of a dataset. It provides a better sense of distribution than the range since it is less influenced by extreme values, also called outliers.
To calculate the IQR, follow these steps:
  • Arrange the data in ascending order (already done: 15, 20, 25, 25, 27, 28, 30, 34).
  • Find the median, which is the average of the 4th and 5th values here, calculated as: Median = \(\frac{(25 + 27)}{2} = 26\).
  • Determine the first quartile (\(Q_1\)), which is the median of the first half of the data: \(Q_1 = 22.5\).
  • Identify the third quartile (\(Q_3\)), which is the median of the second half: \(Q_3 = 29\).
  • Finally, subtract \(Q_1\) from \(Q_3\) to get the IQR: IQR = 29 - 22.5 = 6.5.
This value indicates the range of the middle two quartiles and is useful in understanding the concentration of central data values.
Variance
Variance is a statistical measure that gives insight into the spread of a dataset. It tells us how far each number in the set is from the mean and thus from every other number in the set.
To compute variance:
  • First, determine the mean (average) of the dataset: Mean = 25.5.
  • Next, find the squared difference between each data point and the mean, and then sum these squared differences.
  • The squared differences for each value (15, 20, 25, 25, 27, 28, 30, 34) are respectively: 110.25, 30.25, 0.25, 0.25, 2.25, 6.25, 20.25, and 72.25.
  • Add these squared differences: Total = 242.
  • Finally, divide by the number of data points: Variance = \(\frac{242}{8} = 30.25\).
Variance quantifies variability; higher values signify more spread in the data points from the mean.
Standard Deviation
Standard deviation is like a companion to variance, taking the square root of the variance to return to the same units as the data.
It provides an easy-to-interpret measure of the spread or dispersion of data points around the mean.
  • Since variance for the dataset is 30.25, the standard deviation would be the square root of 30.25.
  • Calculating the square root: Standard Deviation = \(\sqrt{30.25} \approx 5.5\).
A smaller standard deviation illustrates that the data points are generally close to the mean, whereas a larger standard deviation indicates data points spread out over a wider range of values.

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Most popular questions from this chapter

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