The data file Sun contains the volumes for a random sample...

The data file Sun contains the volumes for a random sample of 100 bottles $(237 \mathrm{~mL})$ of a new suntan lotion.
a. Find and interpret the mean volume.
b. Determine the median volume.
c. Are the data symmetric or skewed? Explain.
d. Find the five-number summary for this data.

Key Concepts

Five-number Summary

The five-number summary is a descriptive statistic that includes the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum of a dataset. This summary provides a clear picture of the distribution’s range, spread, and central tendency, and is useful for identifying outliers and understanding the overall variability of the data.

Skewness

Skewness is a measure of the asymmetry of a distribution. In a symmetric distribution, the left and right sides of the histogram are mirror images and the mean and median are approximately equal. In contrast, a skewed distribution indicates that one tail is longer than the other, which is generally evident when the mean and median do not coincide.

Mean

The mean is the arithmetic average of a dataset, obtained by summing all the values and dividing by the number of observations. It provides a single value that represents the central location of the data and is useful for summarizing the overall level of measurements, though it can be influenced by extreme values in the dataset.

Median

The median is the middle value of a dataset when the numbers are arranged in ascending order. It divides the data into two equal halves and is a robust measure of central tendency, especially in the presence of outliers or skewed distributions, because it is not affected by extreme values.

Transcript

00:02 Okay, with this problem, we're going to be thinking about a few different things.

00:07 We're going to be thinking about z scores, quartiles, intercortile range, and outliers.

00:14 And so the context is they give us some data showing the miles per gallon for a random sample of smart cars.

00:27 So here was the data that they gave us.

00:30 And the first thing they ask us to do is to compute the z score for a car that got 36 .3 miles per gallon.

00:44 So for this guy right here.

00:49 So remember a z score is the value itself minus the mean, divided by the same.

01:01 Standard deviation.

01:03 So i needed to find in order to work the problem, i needed to find the mean of the sample and the standard deviation.

01:13 So i put the data into my ti -84 calculator and calculated the one variable stats.

01:19 So i had the information i needed.

01:21 So i'm going to plug in the values 36 .3 minus the mean of 38 .775 divided by the standard deviation, 3 .416.

01:39 And let's go ahead and plug that in, 36 .3 minus 38 .775 divided by 3 .416.

01:50 Gives me a z score of negative 0 .775, gives me a z score of negative 0 .775, gives me a z score of negative so what this means, the interpretation would be that this smart car has an mpg that is 0 .725 standard deviations below the mean.

02:33 Average because it's below.

02:35 And notice when i interpreted it, the negative became below.

02:41 The second thing it asked us to do is to find the quartiles.

02:48 And actually, because i had used my calculator to find the mean and standard deviation, it also gave me the quartiles here.

02:58 Oops, and let me undo that because i have covered it up where i can't see it.

03:04 I also want the median.

03:07 So here's my q1 median, remember, is q2, and q3 is 41...

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