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Chapter 3: Problem 3

The 20 subjects used in Data Set 8 "IQ and Brain Size" in Appendix B have weights with a standard deviation of \(20.0414 \mathrm{kg}\). What is the variance of their weights? Be sure to include the appropriate units with the result.

Short Answer

Expert verified
The variance is 401.6568 \mathrm{kg}^2.

Step by step solution

01

- Understanding Standard Deviation

The standard deviation given is for the weights of 20 subjects, which is denoted as \(\text{SD} = 20.0414 \mathrm{kg}\). The standard deviation measures the dispersion of data points from the mean.
02

- Definition of Variance

Variance measures the average of the squared differences from the mean. It is denoted as \(\text{Var}\). The relationship between variance and standard deviation is given by the formula \(\text{Var} = \text{SD}^2\).
03

- Calculating Variance

Using the standard deviation given, compute the variance: \(\text{Var} = (20.0414 \mathrm{kg})^2 \).
04

- Performing the Calculation

Calculate the square of 20.0414 to get the variance: \( \text{Var} = 20.0414^2 = 401.6568 \mathrm{kg}^2 \).
05

- Result

The variance of the weights of the 20 subjects is \( \text{Var} = 401.6568 \mathrm{kg}^2 \).

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Key Concepts

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

Standard Deviation
Understanding standard deviation is key in statistics. It measures how spread out the numbers in a data set are from the mean.
This concept tells us about the variability or consistency of the data.
For instance, a low standard deviation means the data points are close to the mean.
Conversely, a high standard deviation means the data points are more spread out.
In the given exercise, the standard deviation of the weights is 20.0414 kg.
It indicates that weights deviate from the mean weight by an average of 20.0414 kg.
Variance
Variance is another crucial statistical measure. It reflects the average of the squared differences from the mean.
This gives a broader sense of data variability. The key relation between standard deviation and variance is that variance is the square of the standard deviation.
Mathematically, it is represented as \(\text{Var} = \text{SD}^2\).
In the exercise, to find the variance with a standard deviation of 20.0414 kg, we calculate \(20.0414^2\) which equals 401.6568 \(kg^2\)
Thus, the variance provides a squared unit (like \(kg^2\) for weights) to understand dispersion.
Statistical Calculations
Statistical calculations are essential for interpreting data precisely.
They include methods to find mean, median, standard deviation, variance, etc.
In the given problem, we first identify the standard deviation and use it to find the variance.
These calculations help in making informed decisions based on data. For standard deviation, start by calculating the mean of the dataset.
Then determine the deviations of each data point from the mean, square these deviations, and take their average.
The standard deviation is the square root of this average, while variance skips the final square root step. Accurate statistical calculations are fundamental in various fields such as research, economics, psychology, and more.
Data Analysis
Data analysis involves inspecting, cleaning, and modeling data to extract useful information.
It makes use of statistical tools like standard deviation and variance to understand data trends and patterns.
This process enhances decision-making capabilities. For instance, knowing the variance in weights helps in understanding how consistent the data is.
A low variance implies that the weights are pretty consistent, whereas a high variance indicates more variability.
Effective data analysis combines these statistical measures with proper interpretation.
This enables solving real-world problems, optimizing processes, and predicting future trends based on past data.

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Most popular questions from this chapter

Express all z scores with two decimal places. Pulse rates of adult females are listed in Data Set 1 "Body Data" in Appendix B. The lowest pulse rate is 36 beats per minute, the mean of the listed pulse rates is \(\bar{x}=74.0\) beats per minute, and their standard deviation is \(s=12.5\) beats per minute. a. What is the difference between the pulse rate of 36 beats per minute and the mean pulse rate of the females? b. How many standard deviations is that [the difference found in part (a)]? c. Convert the pulse rate of 36 beats per minutes to a \(z\) score. d. If we consider pulse rates that convert to z scores between -2 and 2 to be neither significantly low nor significantly high, is the pulse rate of 36 beats per minute significant?

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