The following data were reported on oxygen consumption (mL/kg/min) for a sample of ten firefighters performing a fire-suppression simulation. 33.0, 47.9, 27.1, 29.9, 26.5, 28.6, 33... The following data were reported on oxygen consumption (mL/kg/min) for a sample of ten firefighters performing a fire-suppression simulation. 33.0, 47.9, 27.1, 29.9, 26.5, 28.6, 33.3, 29.0, 24.3, 27.2 Compute the following: (a) the sample range (b) the sample variance s2 from the definition (c) the sample standard deviation (d) s2 using the shortcut method.
Understand the Problem
The question is asking to perform various statistical computations based on the provided data set relating to oxygen consumption for firefighters, specifically calculating the sample range, variance, standard deviation and using a shortcut method for variance.
Answer
Sample Range = $X_{max} - X_{min}$; Sample Variance = $s^2$; Sample Standard Deviation = $s$.
Answer for screen readers
The final answer will depend on the values calculated from the data set.
Sample Range = $X_{max} - X_{min}$,
Sample Variance = $s^2$,
Sample Standard Deviation = $s$.
Steps to Solve
- Calculate the Sample Range
To find the sample range, subtract the smallest value from the largest value in the data set.
Let’s define:
- Max value = $X_{max}$
- Min value = $X_{min}$
The formula for the sample range is: $$ \text{Range} = X_{max} - X_{min} $$
- Calculate the Sample Mean
To calculate the sample mean, sum all the values in the data set and divide by the number of values ($n$).
Let:
- $X_1, X_2, ..., X_n$ be the data values
The formula for the sample mean is: $$ \bar{X} = \frac{X_1 + X_2 + ... + X_n}{n} $$
- Calculate the Sample Variance
Using the formula for sample variance, first find the squared differences from the mean, then average those squared differences.
The formula for sample variance ($s^2$) is: $$ s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1} $$
- Calculate the Sample Standard Deviation
The standard deviation is simply the square root of the sample variance.
The formula for sample standard deviation ($s$) is: $$ s = \sqrt{s^2} $$
- Shortcut Method for Variance
The shortcut method involves using the formula: $$ s^2 = \frac{n \sum X_i^2 - (\sum X_i)^2}{n(n-1)} $$
Where:
- $\sum X_i^2$ is the sum of squares of the data values.
- $\sum X_i$ is the sum of the data values.
- $n$ is the number of values in the data set.
The final answer will depend on the values calculated from the data set.
Sample Range = $X_{max} - X_{min}$,
Sample Variance = $s^2$,
Sample Standard Deviation = $s$.
More Information
These statistical measures help assess data variability. The range gives a simple measure of spread, while variance and standard deviation indicate how data points are dispersed around the mean.
Tips
- Forgetting to divide by $n-1$ when calculating sample variance, which leads to an incorrect bias.
- Mixing up the formulas for population variance and sample variance.
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