Use the following sets of numbers. They are the same as those used in
Exercises 22.2.
\(A: 3,6,4,2,5,4,7,6,3,4,6,4,5,7,3\)
\(B: 25,26,23,24,25,28,26,27,23,28,25\)
\(C: 0.48, 0.53, 0.49, 0.45, 0.55, 0.49, 0.47, 0.55, 0.48, 0.57, 0.51,
0.46,0.53,0.50,0.49,0.53\)
\(D:
105,108,103,108,106,104,109,104,110,108,108,104,113,106,107,106,107,109,105,111,
109,108\)
Use \(E q\). (22.3) to find the standard deviation s for the indicated sets of
numbers. Set \(C\)
The standard deviation for set C is approximately 0.0242.
Step by step solution
01
Calculate the Mean of Set C
First, find the mean (average) of the numbers in set C. The formula for the mean \( \bar{x} \) is \( \bar{x} = \frac{\sum x_i}{n} \), where \( \sum x_i \) is the sum of all data points, and \( n \) is the number of data points.For set C: \[ \sum x_i = 0.48 + 0.53 + 0.49 + 0.45 + 0.55 + 0.49 + 0.47 + 0.55 + 0.48 + 0.57 + 0.51 + 0.46 + 0.53 + 0.50 + 0.49 + 0.53 = 7.78 \] And \( n = 16 \). So, \( \bar{x} = \frac{7.78}{16} = 0.48625 \).
02
Calculate the Variance
The variance \( s^2 \) is computed using the formula: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \] Calculate each \((x_i - \bar{x})^2\) for the data points in C:- \((0.48 - 0.48625)^2 = 0.00003906\)- \((0.53 - 0.48625)^2 = 0.00192556\)- Continue similar calculations for each element.Sum these squared differences: \[ \sum (x_i - \bar{x})^2 = 0.00003906 + 0.00192556 + ... + 0.00192556 = 0.0087625 \]The variance \( s^2 = \frac{0.0087625}{15} = 0.0005841667 \).
03
Calculate the Standard Deviation
The standard deviation \( s \) is the square root of the variance:\[ s = \sqrt{s^2} = \sqrt{0.0005841667} \approx 0.0241663 \].
04
Interpret the Result
The standard deviation indicates how spread out the numbers are in set C around their mean. A smaller standard deviation, like \( 0.0241663 \), suggests the numbers are fairly close to the mean.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mean Calculation
Calculating the mean is an essential part of understanding any data set. The mean, often referred to as the average, gives us a central value that represents the data distribution. To calculate the mean, you sum up all the values in the data set and then divide by the total number of values. This process is quite straightforward.
For example, in data set C:
- First, add up all the numbers in the set:
\(0.48 + 0.53 + 0.49 + 0.45 + 0.55 + 0.49 + 0.47 + 0.55 + 0.48 + 0.57 + 0.51 + 0.46 + 0.53 + 0.50 + 0.49 + 0.53 = 7.78\) - Next, divide this sum by the number of data points:
\(n = 16\), so \(\bar{x} = \frac{7.78}{16} = 0.48625\)
The mean tells us that each number in set C is, on average, approximately 0.48625. It's the first step in many statistical analyses.
Variance Calculation
Variance is a measure of how much the numbers in a data set differ from the mean value. It shows the data's variability or spread. Calculating variance is important because it enables us to determine how individual numbers differ from the mean, which is crucial in statistics.
To calculate variance, you follow these steps:
- Subtract the mean from each number to find the deviation of each data point from the mean.
- Square each of these deviations.
- Average these squared deviations, but use \(n-1\) as the divisor if you are dealing with a sample of the data.
Applying this to data set C, we square the deviations and sum them up, then divide by 15 (since \(n=16\)), resulting in a variance of approximately 0.0005841667. This value gives us a quantitative idea of how spread out the data points are about the mean.
Data Sets
A data set is a collection of numbers or values that relate to a particular subject. It forms the basis of analysis in statistics. Each data set can tell us different things, depending on how it's structured and what we're trying to find out.
In our exercise, we deal with several data sets (e.g., sets A, B, C, and D), each with various numerical values. These sets allow us to practice calculating mean, variance, and standard deviation.
Understanding different data sets is vital:
- Set A might represent daily temperatures.
- Set B could be test scores.
- Set C may represent percentages, as shown in this exercise.
Talking about data sets is like talking about the story behind the numbers. By analyzing them, we can extract meaningful patterns and conclusions.
Mathematics Education
Mathematics education is about more than just numbers; it involves teaching students the skills they need to analyze and interpret numerical data. Understanding statistics, such as mean, variance, and standard deviation, is crucial in many fields of study.
In educational settings:
- These statistics help students learn to summarize large sets of data effectively.
- They also aid in critical thinking, helping students predict outcomes and trends.
- By mastering these concepts, students are better prepared to handle real-world data.
Educators focus on simplifying these concepts with exercises and examples, like the one we analyzed, to make learning engaging and accessible. By doing so, they foster a deeper understanding and appreciation of mathematics and its applications.
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