Determine the range and sample standard deviation for each...

Determine the range and sample standard deviation for each of the data sets. For the sample standard deviation, round each answer to one more decimal place than that used for the observations. Every year, Consumer Reports publishes a magazine titled New Car Ratings and Review that looks at vehicle profiles for the year's models. It lets you see in one place how, within each category, the vehicles compare. One category of interest, especially when fuel prices are rising, is fuel economy, measured in miles per gallon (mpg). Following is a list of overall mpg for 14 different full-sized and compact pickups. $$\begin{array}{ccccccc} \hline 14 & 13 & 14 & 13 & 14 & 14 & 11 \\ 12 & 15 & 15 & 17 & 14 & 15 & 16 \\ \hline \end{array}$$

Determine the range and sample standard deviation for each of the data sets. For the sample standard deviation, round each answer to one more decimal place than that used for the observations.
Every year, Consumer Reports publishes a magazine titled New Car Ratings and Review that looks at vehicle profiles for the year's models. It lets you see in one place how, within each category, the vehicles compare. One category of interest, especially when fuel prices are rising, is fuel economy, measured in miles per gallon (mpg). Following is a list of overall mpg for 14 different full-sized and compact pickups.
$$\begin{array}{ccccccc}
\hline 14 & 13 & 14 & 13 & 14 & 14 & 11 \\
12 & 15 & 15 & 17 & 14 & 15 & 16 \\
\hline
\end{array}$$

Key Concepts

Range

Range is a basic measure of spread or dispersion in a data set, calculated as the difference between the maximum and minimum values. It gives a quick sense of how spread apart the extremes in the data are, but does not reflect variations among the remaining data points.

Sample Standard Deviation

Sample standard deviation quantifies the amount of variation or dispersion of a set of sample values around the sample mean. It is computed by taking the square root of the average of the squared deviations from the mean, using n – 1 (degrees of freedom) in the denominator to ensure that the estimate is unbiased when generalizing from a sample to a population.

Rounding Precision

Rounding precision in this context refers to the practice of expressing the calculated statistical measures to a consistent number of decimal places, typically one more than that used in the original observations, to maintain the appropriate level of precision and reflect measurement accuracy.

Transcript

0:00 All right.

00:01 So for this question, we are given a number of car ratings and we are for mpg, miles per gallon for 14 different size cars.

00:10 And we want to find the range in the sample standard deviation for this data set.

00:16 And all i've actually done here is i've sorted the data from smallest to largest.

00:21 This does two things.

00:22 Firstly, it allows us to group similar terms.

00:25 And secondly, for finding the range, it becomes very straightforward because the range is, you know, simply the biggest minus the smallest so we can very quickly and easily see it at the biggest 17 the smallest is 11 which gives us a range of six all right now when we're trying to find the mean well the mean relies on the sum divided by the total number of terms so add them all up and divide by the total which is 14 why do we need to know the mean though right the problem is asking for the standard deviation well the standard deviation relies on on the squared differences from the mean.

01:04 So if we're going to find the standard deviation, we need to first find the mean.

01:09 So let's add it all up, divide by 14.

01:12 That's going to give us a mean of 14 .1 miles per gallon, which makes sense, right? it's right here in the middle.

01:20 It's very close to 14, actually.

01:23 All right.

01:24 So when we are finding the sample standard deviation, it's a square root expression where the numerator is the sum of the squared differences from the means.

01:35 So what does that mean? it means that for each of our terms, so start with 11, work the way up to 18, we are going to find the squared difference from the mean.

01:46 So subtract the mean, square it.

01:50 Add the next one, right? subtrap the mean, square it.

01:55 And actually notice, for several of our terms here, so if 13 is copied twice, we can just do two times 13 minus the mean squared because we're going to have two copies of those.

02:09 And actually 14 is listed five times.

02:12 So we can do five times 14 minus 14 .1 squared...

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