/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Consider the data set \(\begin... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 6

Consider the data set \(\begin{array}{lllll}1 & 2 & 3 & 4 & 5\end{array}\) (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
Range: 4; Sample standard deviation: 1.58; Population standard deviation: 1.41

Step by step solution

01

Find the Range

To find the range of a data set, we subtract the smallest value from the largest value. For this data set, the smallest value is 1 and the largest value is 5. So,\[\text{Range} = 5 - 1 = 4\]
02

Calculate the Sample Mean

The sample mean \( \bar{x} \) is obtained by summing all data points and then dividing by the number of data points. The sum of the data points is \( 1 + 2 + 3 + 4 + 5 = 15 \), and the number of data points is 5. Thus,\[\bar{x} = \frac{15}{5} = 3\]
03

Compute the Sample Variance

Sample variance \( s^2 \) is calculated using the formula \( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \). First, compute each \((x_i - \bar{x})^2\):\((1-3)^2 = 4\), \((2-3)^2 = 1\), \((3-3)^2 = 0\), \((4-3)^2 = 1\), \((5-3)^2 = 4\).Sum: \(4 + 1 + 0 + 1 + 4 = 10\).Thus,\[s^2 = \frac{10}{4} = 2.5\]
04

Find the Sample Standard Deviation

The sample standard deviation \( s \) is the square root of the sample variance. Therefore,\[s = \sqrt{2.5} \approx 1.58\]
05

Compute the Population Variance

Population variance \( \sigma^2 \) uses the formula \( \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \), where \( \mu \) is the population mean, equal to the sample mean \( \mu = 3 \).We already calculated \( \sum_{i=1}^{n} (x_i - \mu)^2 = 10 \).Thus,\[\sigma^2 = \frac{10}{5} = 2\]
06

Find the Population Standard Deviation

The population standard deviation \( \sigma \) is the square root of the population variance. Therefore,\[\sigma = \sqrt{2} \approx 1.41\]

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Key Concepts

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

Understanding the Range
The range is the simplest way to measure the spread of data in a dataset. It tells us how much distance there is between the smallest and largest values. Knowing the range helps to get a quick idea of how varied our data is. To calculate the range, subtract the smallest number in your dataset from the largest. For our example data set \(1, 2, 3, 4, 5\), the smallest value is 1 and the largest is 5. By calculating \(5 - 1 = 4\), we find that the range is 4.
This means the data spreads across four units from end to end. Remember, while the range gives a general sense of data spread, it doesn鈥檛 show how values are distributed in between.
Delving into Standard Deviation
Standard deviation is a key concept in statistics that tells us how much individual data points differ from the mean of the dataset. In simpler terms, it's a measure of how spread out the numbers are. There are two types of standard deviation: sample and population.
  • Sample Standard Deviation (\(s\)): This is used when working with a sample from a larger population. To find it, first calculate the sample variance, then take the square root of that variance. For our example, we found that \(s^2 = 2.5\) leading to \(s = \sqrt{2.5}\) which approximately equals 1.58.
  • Population Standard Deviation (\(\sigma\)): This applies when your dataset includes the entire population. Similar to the sample standard deviation, it's the square root of the population variance. We computed \(\sigma^2 = 2\), hence \(\sigma = \sqrt{2}\) approximately equals 1.41.
The closer the data points are to the mean, the smaller the standard deviation. Conversely, more spread data results in a higher standard deviation.
Understanding Variance
Variance measures the average of the squared differences from the mean. It provides an idea of how data points disperse around the mean and is a foundational concept for calculating standard deviation.
  • Sample Variance (\(s^2\)): Calculated by taking each data point's deviation from the sample mean, squaring it, summing the squared differences, and then dividing by \(n-1\). For our data, this was calculated as \((10/4 = 2.5)\).
  • Population Variance (\(\sigma^2\)): Similar to sample variance but divided by \(n\) when considering the entire population. We found \[\sigma^2 = \frac{10}{5} = 2\].
Variance is expressed in squared units, which can sometimes make interpretation tricky as it's not in the same units as the data. This is reason we use the square root of variance to get the standard deviation.
Decoding the Population Mean
The population mean, represented by \(\mu\), is the arithmetic average of all data points in a population. It's a central value that summarizes a data set into a single representative value. Calculating the population mean is straightforward: sum all the numbers and divide by how many numbers there are.
The dataset example \((1+2+3+4+5=15)\) divided by 5 gives us \(\mu = 3\).
This number, the population mean, acts as a balance point of the dataset.
While the mean is a useful measure, it doesn't capture data variability. It's helpful to pair the mean with measures such as standard deviation to gain deeper insights into data behavior.

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