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Chapter 3: Problem 5

Consider the data set $$ \begin{array}{lllll} 2 & 3 & 4 & 5 & 6 \end{array} $$ (a) Find the range. (b) Use the defining formula to compute the sample standard deviation \(s\). (c) Use the defining formula to compute the population standard deviation \(\sigma\).

Short Answer

Expert verified
The range is 4. The sample standard deviation is approximately 1.58, and the population standard deviation is approximately 1.41.

Step by step solution

01

Determine the Range

The range of a data set is the difference between the maximum and minimum values. Here, the maximum value is 6, and the minimum value is 2. Thus, the range is calculated as follows: \[ \text{Range} = 6 - 2 = 4 \]
02

Compute the Sample Mean

To find the sample standard deviation, first find the sample mean. Sum the data values and divide by the number of data points, which is 5. \[ \bar{x} = \frac{2 + 3 + 4 + 5 + 6}{5} = \frac{20}{5} = 4 \]
03

Compute Each Deviation from the Mean and Square It

Calculate the deviation of each data point from the sample mean, and then square each deviation. \[(2-4)^2 = 4, \quad (3-4)^2 = 1, \quad (4-4)^2 = 0, \quad (5-4)^2 = 1, \quad (6-4)^2 = 4\]
04

Find the Sample Variance

Sum the squared deviations and divide by \(n-1\), where \(n=5\) is the number of data points. \[ s^2 = \frac{4 + 1 + 0 + 1 + 4}{5-1} = \frac{10}{4} = 2.5 \]
05

Find the Sample Standard Deviation

Take the square root of the sample variance to find the sample standard deviation. \[ s = \sqrt{2.5} \approx 1.58 \]
06

Compute the Population Variance

For the population standard deviation, sum the squared deviations and divide by \(n\). \[ \sigma^2 = \frac{4 + 1 + 0 + 1 + 4}{5} = \frac{10}{5} = 2 \]
07

Find the Population Standard Deviation

Take the square root of the population variance to find the population standard deviation. \[ \sigma = \sqrt{2} \approx 1.41 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Range in Statistics
The range is a simple measure of variability in a data set. It gives us a basic sense of how spread out the data values are. You can find it by subtracting the smallest data value from the largest one.
For example, with the data set \( \{2, 3, 4, 5, 6\} \), the range is calculated as follows:
  • Maximum value: 6
  • Minimum value: 2
  • Range: \( 6 - 2 = 4 \)
The range provides a quick glimpse of the spread but doesn鈥檛 tell us about the distribution of other data points in the set. Therefore, it is often combined with other measures, such as variance and standard deviation, for a thorough analysis.
Understanding Standard Deviation
Standard deviation is a more comprehensive measure of variability than range. It tells us how much individual data points differ from the mean.
To find the standard deviation, you first need to determine the mean (average) of the dataset. For our dataset, we have:
  • Sum of data points: \(2 + 3 + 4 + 5 + 6 = 20\)
  • Number of data points: 5
  • Sample Mean, \( \bar{x} = \frac{20}{5} = 4 \)
Then, find the difference between each data point and the mean, square these differences, and use them in calculating variance (a precursor to standard deviation). Finishing with the square root gives us the standard deviation, which reflects data dispersion.
The standard deviation is crucial because it reflects the average distance of each data point from the mean, allowing us to understand how "spread out" the data is.
Sample Variance: A Closer Look
Sample variance provides a key insight into data variability, helping us understand the degree to which data points in a sample vary from the mean.
To compute the sample variance:
  • Calculate each data point's deviation from the sample mean and square it.
  • Sum these squared deviations.
  • Divide the total by the number of data points minus one \((n - 1)\). This operation corrects the bias in the estimation of population variance from a sample.
For our data set, the sample variance \( s^2 \) is calculated as:\[ s^2 = \frac{4 + 1 + 0 + 1 + 4}{5-1} = 2.5\]This measure is essential, especially when working with a sample rather than the whole population, because it provides a better estimate of the overall variability.
Exploring Population Variance
Population variance is quite similar to sample variance, yet it's computed using all data points in the entire population rather than a portion of it.
To find the population variance:
  • Calculate the deviation of each data point from the population mean.
  • Square these deviations and sum them.
  • Divide the total by the number of data points \(n\).
In our example with population variance \( \sigma^2 \):\[\sigma^2 = \frac{4 + 1 + 0 + 1 + 4}{5} = 2\]The resulting value indicates the average of the squared deviations from the mean and is useful to understand how much the data can deviate from the average within a population. Unlike sample variance, it's unbiased because it considers every single data point.

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