The following data represent the age of U.S. presidents on...

The following data represent the age of U.S. presidents on their respective inauguration days (through Barack Obama). $$\begin{equation}\begin{array}{lllllllll}42 & 47 & 50 & 52 & 54 & 55 & 57 & 61 & 64 \\\hline 43 & 48 & 51 & 52 & 54 & 56 & 57 & 61 & 65 \\\hline 46 & 49 & 51 & 54 & 55 & 56 & 57 & 61 & 68 \\\hline 46 & 49 & 51 & 54 & 55 & 56 & 58 & 62 & 69 \\\hline 47 & 50 & 51 & 54 & 55 & 57 & 60 & 64 & \\\hline\end{array}\end{equation}$$ (a) Find the five-number summary. (b) Construct a boxplot. (c) Comment on the shape of the distribution.

The following data represent the age of U.S. presidents on their respective inauguration days (through Barack Obama).
$$\begin{equation}\begin{array}{lllllllll}42 & 47 & 50 & 52 & 54 & 55 & 57 & 61 & 64 \\\hline 43 & 48 & 51 & 52 & 54 & 56 & 57 & 61 & 65 \\\hline 46 & 49 & 51 & 54 & 55 & 56 & 57 & 61 & 68 \\\hline 46 & 49 & 51 & 54 & 55 & 56 & 58 & 62 & 69 \\\hline 47 & 50 & 51 & 54 & 55 & 57 & 60 & 64 & \\\hline\end{array}\end{equation}$$
(a) Find the five-number summary.
(b) Construct a boxplot.
(c) Comment on the shape of the distribution.

Key Concepts

Distribution shape

The shape of the distribution refers to how the data values are spread and can indicate skewness (whether the data are more spread out on one side), symmetry, or the presence of outliers. By examining the relative positions of the median and quartiles, as well as the lengths of the whiskers in a boxplot, one can comment on whether the distribution is symmetric, positively skewed, or negatively skewed.

Five-number summary

The five-number summary consists of five descriptive statistics: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. It provides a concise description of the distribution of a data set by highlighting its central tendency, spread, and potential outliers while summarizing the entire range of the data.

Boxplot

A boxplot is a graphical representation of the five-number summary that visually displays the distribution of a data set. It features a box stretching from Q1 to Q3 with a line at the median. The 'whiskers' extend from the box to the minimum and maximum values (or to defined bounds beyond which values are considered outliers), providing insight into the spread and skewness of the data.

Transcript

00:01 All right.

00:01 With this question, we're given a set of data that represent the ages of president at their inauguration up through obama.

00:13 And we're asked to do a couple things.

00:16 They want us to come up with the five number summary, turn it into a box plot, and then describe the shape.

00:22 So we're going to start with the five number summary.

00:26 The minimum is always easy, just the lower.

00:31 Lowest value, so the minimum is 42.

00:34 The maximum, also easy to find.

00:37 It's just the largest value, so 69.

00:40 Then we need to find the median.

00:42 Now here, if you notice there's five values, so if you count them up, there's 44 values total.

00:49 So the median is going to be between the 22nd and 23rd, the average of those two.

00:57 So 10, 20, 22, 23.

00:59 So 10, 20, 22, 23.

01:01 In between here.

01:03 So 54 and 55, the median is going to be 54 .5, halfway between those two numbers.

01:11 And then for q1, with q1, what we're going to do is we want the middle of the bottom 22 values, so the middle of these values.

01:26 So that's going to be between the 11th and the 12th.

01:29 So it's going to be right here.

01:32 And halfway between 50 and 51 is 50 .5.

01:38 And then for q3, we want the middle of the upper half of the data, and that's going to be between the 11th and 12th terms here.

01:53 It's going to be right there.

01:55 Let's see.

01:56 One, two, three, four, five, six, seven, eight, nine, ten, eleven.

02:00 Yeah.

02:00 So that would give me halfway between 57 and 58 is 57 .5.

02:06 So there's my five number summary.

02:09 Before i can actually do the box plot, i need to decide if we have outliers...

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