Calculate [latex]z[/latex]-scores to explain the location of data points.
Compare observations using [latex]z[/latex]-scores and the Empirical Rule.
Recall, to calculate a [latex]z[/latex]-score given an observation, use the formula [latex]z=\dfrac{x-\mu}{\sigma}[/latex], where [latex]x[/latex] represents the value of the observation, [latex]\mu[/latex] represents the population mean, [latex]\sigma[/latex] represents the population’s standard deviation, and [latex]z[/latex] represents the standardized value, or z-score.Let’s consider the data set Sleep Study: Average Sleep. Let’s calculate [latex]z[/latex]-scores for individual observations in the data set. The mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex] in the formula represent the population the sample came from.
Since we don’t know these, we’ll use the sample mean and standard deviation in our calculations. The mean of the data is [latex]7.97[/latex] hours with a standard deviation of [latex]0.965[/latex]. Calculate the [latex]z[/latex]-scores for each of the following observations and indicate if the given value lies above or below the mean. Round your calculations to two decimal places.
[latex]6.93\text{ hours}[/latex]
[latex]9.87\text{ hours}[/latex]
[latex]7.97\text{ hours}[/latex]
[latex]4.95\text{ hours}[/latex]
[latex]z=\dfrac{6.93-7.97}{0.965}\approx -1.08[/latex]. This value is [latex]1.08[/latex] standard deviations below the mean.
[latex]z=\dfrac{9.87-7.97}{0.965}\approx 1.97[/latex]. This value is [latex]1.97[/latex] standard deviations above the mean.
[latex]z=\dfrac{7.97-7.97}{0.965}=0[/latex]. This value is equal to the mean.
[latex]z=\dfrac{4.95-7.97}{0.965}\approx -3.13[/latex]. This value is [latex]-3.13[/latex] standard deviations below the mean.
Let’s look at the Sleep Study: Average Sleep data set and learn about standard deviation as a measure of the variability of a data set. Use the statistical tool above to select the Sleep Study: Average Sleep data set.
Display a histogram and dotplot and make a note of the mean and standard deviation in the descriptive statistics. Round your final answers to the questions below to 3 decimal places, as needed.
Describe the shape of the data set using the histogram and dotplot. For practice, display a boxplot as well and note the visual clues that you can use to determine the shape of the distribution from the boxplot.
How does the relationship between the mean and median (given in descriptive statistics) help to support your analysis?
What are the mean and standard deviation of the data set?
What number of sleep hours lies one standard deviation above the mean? What value lies one standard deviation below?
What number of sleep hours lie two standard deviations above and below the mean?
The distribution is unimodal and approximately symmetric. A few outliers lie evenly to either side of the distribution.
The mean and median are approximately equal, which supports that the outliers are approximately evenly distributed.
[latex]\bar{x}=7.97[/latex] and [latex]s=0.965[/latex].
The standard deviation is 0.965. We can add that to the mean to determine the value exactly one standard deviation above the mean. Likewise, we can subtract that from the mean to find the value exactly one standard deviation below the mean.
[latex]7.97 + 0.965 = 8.935[/latex]:
[latex]8.935[/latex] hours of sleep lies one standard deviation above the mean.
That is, the standardized value for the observation [latex]8.935[/latex] hours is [latex]1[/latex[]. Its [latex]z[/latex]-score is [latex]1[/latex].
[latex]7.97 - 0.965 = 7.005[/latex]:
[latex]7.005[/latex] hours of sleep lies one standard deviation below the mean.
That is, the standardized value for the observation [latex]7.005[/latex] hours is [latex]-1[/latex]. Its [latex]z[/latex]-score is [latex]-1[/latex]
The standard deviation is [latex]0.965[/latex]. We can add twice that to the mean to determine the value exactly two standard deviations above the mean. Likewise, we can subtract [latex]2*0.965[/latex] from the mean to find the value exactly one standard deviation below the mean.
[latex]7.97 + 2*0.965 = 9.9[/latex]:
[latex]9.9[/latex] hours of sleep lies two standard deviations above the mean.
That is, the standardized value for the observation [latex]9.9[/latex] hours is [latex]2[/latex[]. Its [latex]z[/latex]-score is [latex]2[/latex].
[latex]7.97 - 2*0.965 = 6.04[/latex]:
[latex]6.04[/latex] hours of sleep lies two standard deviations below the mean.
That is, the standardized value for the observation [latex]6.04[/latex] hours is [latex]-2[/latex]. Its [latex]z[/latex]-score is [latex]-2[/latex].