Find the range, variance, and standard deviation for the...

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-2 where we found measures of center. Here use find measures of variation.) Then answer the given questions. Listed below are the lengths of time (in years) it took for a random sample of college students to cam bachelor's degrees (based on data from the U.S. National Center for Education Statistics). Based on these results, is it unusual for someone to earn a bachelor's degree in 12 years? $$\begin{array}{rrrrrrrrrrrrrr} 4 & 4 & 4 & 4 & 4 & 4 & 4.5 & 4.5 & 4.5 & 4.5 & 4.5 & 4.5 & 6 & 6 & 8 & 9 & 9 & 13 & 13 & 15 \end{array}$$

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-2 where we found measures of center. Here use find measures of variation.) Then answer the given questions.

Listed below are the lengths of time (in years) it took for a random sample of college students to cam bachelor's degrees (based on data from the U.S. National Center for Education Statistics). Based on these results, is it unusual for someone to earn a bachelor's degree in 12 years?
$$\begin{array}{rrrrrrrrrrrrrr}
4 & 4 & 4 & 4 & 4 & 4 & 4.5 & 4.5 & 4.5 & 4.5 & 4.5 & 4.5 & 6 & 6 & 8 & 9 & 9 & 13 & 13 & 15
\end{array}$$

Key Concepts

Unusual Observations

Determining whether an observation is unusual involves comparing it to the overall distribution of the data, often using the standard deviation. An observation that lies several standard deviations away from the mean may be considered an outlier or an unusual value, providing context for assessing the rarity of specific data points.

Standard Deviation

The standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the data. It illustrates the typical distance that the data points are from the mean, making it a useful indicator of variability and a tool to assess how unusual or typical an individual data value might be.

Variance

Variance quantifies how much the data values differ from the mean by taking the average of the squared differences. In the context of a sample, it is computed by summing the squared differences from the sample mean and dividing by one less than the number of observations (n - 1), thus measuring the spread of the data in squared units.

Range

The range is a basic measure of dispersion that indicates the spread of data by subtracting the smallest value from the largest value in a dataset. It provides a quick sense of the overall spread but can be sensitive to outliers.

Transcript

00:01 In this problem, we have a dataset about the amount of years necessary to complete a bachelor's degree.

00:08 And we want to know if it's unusual to finish a bachelor's in how many years? in 12 years.

00:16 So is it unusual to complete a bachelor's in 12 years based on that sample dataset? so first let's find out how many data points we have.

00:26 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 20.

00:45 So we have 20 data points.

00:49 First, find the average.

00:52 So 1 divided by 20.

00:54 We add all of our data points, so 4 plus 4 plus 4 plus.

00:59 4 plus 4 plus 4 plus 4 .5, blah, blah, add a reading up until you're at 15.

01:06 That will give an average of 6 .5.

01:11 So on average, based on that sample, it's 6 .5 years to complete a bachelor degree.

01:19 Range.

01:22 Now, what's the biggest value? and what's the minimum value? substract the minimum value to the largest value, and that will give us the range.

01:32 So our maximum value is 15.

01:34 That's always good when our data is already sorted and our minimum value is 4.

01:40 So the range will be 15 minus 4, which is 11 years.

01:48 So our sample spends 11 years.

01:52 Now the standard deviation.

01:55 So we use the formula for the sample, so 1 divided by s minus 1, and then the sum of every data point to which we subscribed the mean and square everything...

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