An online pharmaceutical company obtained the following...

An online pharmaceutical company obtained the following frequency distribution of shipping times (number of hours between the time an order is placed and the time the order is shipped) for a random sample of 40 orders. (Be sure to complete all appropriate columns and show your work). $$ \begin{array}{rr} \hline \text { Number of Hours } & f_i \\ \hline 4<10 & 8 \\ 10<16 & 15 \\ 16<22 & 10 \\ 22<28 & 7 \\ \hline \end{array} $$ a. What is the approximate mean shipping time? b. What is the approximate variance and standard deviation?

An online pharmaceutical company obtained the following frequency distribution of shipping times (number of hours between the time an order is placed and the time the order is shipped) for a random sample of 40 orders. (Be sure to complete all appropriate columns and show your work).
$$
\begin{array}{rr}
\hline \text { Number of Hours } & f_i \\
\hline 4<10 & 8 \\
10<16 & 15 \\
16<22 & 10 \\
22<28 & 7 \\
\hline
\end{array}
$$
a. What is the approximate mean shipping time?
b. What is the approximate variance and standard deviation?

Key Concepts

Frequency Distribution

A frequency distribution is a tabular summary that displays the number of observations (or frequency) which fall within specified intervals or bins. It is fundamental in organizing and summarizing large datasets to highlight patterns or trends, such as the distribution of shipping times in this example.

Grouped Data

Grouped data refers to data that has been organized into groups or intervals, rather than being presented as individual data points. In such cases, calculations like the mean and variance must use representative values (usually the midpoints) for each group to approximate the true statistics.

Class Midpoint

The class midpoint is the average of the lower and upper boundaries of an interval in grouped data. It is used as an estimate for all the data points within that interval when computing descriptive statistics such as the mean and variance.

Weighted Mean

The weighted mean in the context of grouped data is calculated by multiplying each class midpoint by its corresponding frequency, summing these products, and then dividing by the total frequency. This method accommodates the varying number of observations in each interval.

Variance

Variance is a measure of the spread or dispersion of a set of data values around the mean. For grouped data, it is estimated by using the class midpoints and frequencies to calculate the squared deviations from the mean, reflecting how the shipping times vary.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of spread in the same units as the original data. It offers insight into the variability of the shipping times and is often used to assess consistency or reliability of the process.

Transcript

00:01 We have to find a mean, a variance, and the standard deviation for this.

00:03 And this was the number of hours of shipping time, and this was the frequency of these.

00:10 And we had, i will say this is really kind of an unusual setting, where it shows four less than 10, and then 10 less than 16.

00:18 And i don't like this notation whatsoever.

00:21 I wish they said 4 is less than x is less than 10 or something like that, where they put a variable in there because kind of an, odd way to show this.

00:33 Anyway, it is what it is and we'll just deal with it.

00:37 So we have a total here for that column.

00:40 They tell us that that total is 40.

00:42 I'm going to take their word for it.

00:44 And we want to find what the midpoint is of each of these columns and of each of these categories.

00:53 And so halfway between these or add them together and divide by two.

00:57 So we get that middle is seven.

00:59 And then the next one is 13, and the next one is 19, and the next one is 25.

01:07 And so we want to find in part a, the mean.

01:10 And so again, this is an approximation for what the mean is, because we don't know what all these eight values have as far as the amount of shipping time, but we're going to take the value of seven, and it's going to occur eight times.

01:23 And we're going to take the value of 13, and it occurs 15 times.

01:27 And we have that value of 19, it occurs 10, and we have that value of 25 that occurs seven times.

01:35 And so we are going to add those all together and then divide by 40, because there are 40 different values.

01:43 And so let me quick get that for you.

01:47 And we find that that mean value comes out to be approximately 15 .4.

01:54 So now we need to find the variance in the standard deviation.

01:57 So let's find the variance first.

01:59 So let's do that in red.

02:01 So the variance, it's a sample variance, and that is an s squared.

02:06 And we have to take each of these values.

02:09 And so we're going to use our midpoint of seven and seven minus and find the deviation from the mean.

02:16 And so that looks like that is negative 8 .4, but i'm just going to write it as 8 .4 because we know we're going to square it.