A. obtain and interpret the quartiles. b. determine and...

a. obtain and interpret the quartiles. b. determine and interpret the interquartile range. c. find and interpret the five-number summary. d. identify potential outliers, if any. e. construct and interpret a boxplot. The U.S. National Center for Health Statistics compiles data on the length of stay by patients in shortterm hospitals and publishes its findings in Vital and Health Statistics. A random sample of 21 patients yielded the following data on length of stay, in days. $$\begin{array}{rrrrrrr} \hline 4 & 4 & 12 & 18 & 9 & 6 & 12 \\ 3 & 6 & 15 & 7 & 3 & 55 & 1 \\ 10 & 13 & 5 & 7 & 1 & 23 & 9 \\ \hline \end{array}$$

a. obtain and interpret the quartiles.
b. determine and interpret the interquartile range.
c. find and interpret the five-number summary.
d. identify potential outliers, if any.
e. construct and interpret a boxplot.
The U.S. National Center for Health Statistics compiles data on the length of stay by patients in shortterm hospitals and publishes its findings in Vital and Health Statistics. A random sample of 21 patients yielded the following data on length of stay, in days.
$$\begin{array}{rrrrrrr}
\hline 4 & 4 & 12 & 18 & 9 & 6 & 12 \\
3 & 6 & 15 & 7 & 3 & 55 & 1 \\
10 & 13 & 5 & 7 & 1 & 23 & 9 \\
\hline
\end{array}$$

Key Concepts

Boxplot

A boxplot, or box-and-whisker plot, is a graphical representation of a data set’s five-number summary. It visually displays the center, spread, and symmetry of the data along with any potential outliers. The box represents the interquartile range with a line at the median, while the whiskers extend to the smallest and largest values within a specified range, making it easier to identify unusual observations.

Quartiles

Quartiles are values that divide a data set into four equal parts after it has been arranged in order. They provide a way to understand the distribution of data by identifying the boundaries where the data is split into four segments, with the first quartile (Q1) marking the 25th percentile, the second quartile (Q2 or median) the 50th percentile, and the third quartile (Q3) the 75th percentile.

Interquartile Range

The interquartile range (IQR) is a measure of variability that represents the range within which the middle 50% of the data fall. It is calculated by subtracting the first quartile from the third quartile (IQR = Q3 - Q1) and is useful for understanding the spread of the data while minimizing the effects of extreme values or outliers.

Five-Number Summary

The five-number summary is a set of descriptive statistics that provide a quick overview of a data set. It consists of the minimum value, first quartile (Q1), median (Q2), third quartile (Q3), and maximum value. This summary helps in understanding the distribution, central tendency, and spread of the data.

Outliers

Outliers are data points that differ significantly from other observations in a dataset. They can indicate variability in the measurement, experimental errors, or a novel discovery. In descriptive statistics, outliers are often identified using the IQR, where any point falling below Q1 - 1.5*IQR or above Q3 + 1.5*IQR is considered an outlier.

Transcript

00:01 Hi everyone, in this video, we're going to use a five number summary and a box block to look at the statistics of 21 patients.

00:09 First, make sure to sort all your data from lowest to highest from 1 to 55.

00:16 Next, we'll find the median or the middle number.

00:20 Because we have 21 patients or 21 observations, we'll look for the 11th patient.

00:27 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

00:40 So this 7 represents our median.

00:43 So write that down.

00:44 Q2.

00:50 Q2, our median is 7.

00:56 Next, we'll find quartile 1, which is half of the first half.

01:02 So make sure to include the median when you're counting for the first half.

01:09 So we have 10 on the left side plus the median.

01:13 So we actually have 11 observations.

01:17 So the middle of 11 is 6, the 6 position or the 6 patient.

01:24 1, 2, 3, 4, 5, 6.

01:29 So the second 4 represents quartal 1.

01:45 Next to find quartal 3, we do the same thing, but also keep in mind to include the median.

01:51 So, 1, 2, 3, 4, 5, 6.

02:00 This 12 will represent quartile 3.

02:10 Next, to interpret these quartiles, we can treat each section of our observation.

02:17 This one section, another section, third section, and fourth section.

02:21 We can treat each section as 25 % of our data.

02:26 So, for example, all the patients to have stayed from 1 to 4 days represent.

02:32 25 % of our data.

02:34 Patients that stay from one day all the way to seven days represents 50 % of our data.

02:41 Likewise, quartile 3, patients have stayed from 1 to 12 days represent 75 % of our data...

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