A survey was conducted of 130 purchasers of new BMW 3 series...

A survey was conducted of 130 purchasers of new BMW 3 series cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of new BMW 7 series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results. (FIGURE CAN'T COPY) a. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series. b. Which group is most likely to have an outlier? Explain how you determined that. c. Compare the three box plots. What do they imply about the age of purchasing a BMW from the series when compared to each other? d. Look at the BMW 5 series. Which quarter has the smallest spread of data? What is the spread? e. Look at the BMW 5 series. Which quarter has the largest spread of data? What is the spread? f. Look at the BMW 5 series. Estimate the interquartile range (IQR). g. Look at the BMW 5 series. Are there more data in the interval 31 to 38 or in the interval 45 to 55? How do you know this? h. Look at the BMW 5 series. Which interval has the fewest data in it? How do you know this? $$\begin{array}{l}{\text { i. } 31-35} \\ {\text { ii. } 38-41} \\ {\text { iii. } 41-64}\end{array}$$

A survey was conducted of 130 purchasers of new BMW 3 series cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of new BMW 7 series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.
(FIGURE CAN'T COPY)
a. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series.
b. Which group is most likely to have an outlier? Explain how you determined that.
c. Compare the three box plots. What do they imply about the age of purchasing a BMW from the series when
compared to each other?
d. Look at the BMW 5 series. Which quarter has the smallest spread of data? What is the spread?
e. Look at the BMW 5 series. Which quarter has the largest spread of data? What is the spread?
f. Look at the BMW 5 series. Estimate the interquartile range (IQR).
g. Look at the BMW 5 series. Are there more data in the interval 31 to 38 or in the interval 45 to 55? How do you know this?
h. Look at the BMW 5 series. Which interval has the fewest data in it? How do you know this?
$$\begin{array}{l}{\text { i. } 31-35} \\ {\text { ii. } 38-41} \\ {\text { iii. } 41-64}\end{array}$$

Key Concepts

Comparative Distribution Analysis

Comparing multiple box plots allows for a visual assessment of differences in central tendency, variability, and data spread across different groups. By examining the box plots side by side, one can infer which groups have higher or lower medians, greater or lesser variability, and whether any group includes unusual or extreme data points.

Interquartile Range (IQR)

The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset. It measures the spread of the middle 50% of the data, offering a robust indicator of variability that is less influenced by extreme values or outliers, compared to the full range.

Spread of Data

The spread of data refers to the extent to which data values are dispersed across the range of the dataset. In a box plot, this is observed through the overall range (from minimum to maximum) as well as through the interquartile range (the spread of the middle 50% of the data). Understanding the data spread is important for assessing consistency and variability within a dataset.

Data Interval Frequency

Analyzing the frequency of data within specific intervals involves looking at how data points are distributed across segmented ranges of values. This concept is related to understanding density and concentration of data in different parts of the distribution, which can be inferred from the width of the quartiles or other intervals on the box plot.

Distribution Shape

The shape of a distribution, as depicted in a box plot, can indicate symmetry or skewness. A symmetric box plot shows the median centrally located with roughly equal lengths for the whiskers and boxes on either side, while a skewed plot will have one side stretched out longer than the other, reflecting a concentration of values toward one end and a tail on the opposite side.

Box Plot

A box plot, or box-and-whisker plot, is a statistical graphic that displays the distribution of a dataset by illustrating its minimum, first quartile, median, third quartile, and maximum. It provides a concise visual summary that highlights the central tendency, variability, and potential outliers within the data, making it useful for comparing several groups simultaneously.

Outliers

Outliers are data values that lie far beyond the typical range of the rest of the data. In a box plot, they are often marked as individual points beyond 1.5 times the interquartile range from the quartile boundaries. Identifying outliers is crucial because they can affect the interpretation of data, indicate anomalies, or suggest the presence of variability not captured by the standard spread measures.

Transcript

00:02 Okay, for this problem, we're comparing some box plots and i'll have you refer to your book for a little cleaner version, but i'll do my best to my little sketch there of the three different box plots we're going to compare so let's take a look and for part a i wants to describe the shape okay, so let's go for part a in shape so we're looking at the shape, what they mean by shape is they really want to see the shape or the spread as they're looking at so if i look at the top orange ones a bmw three series.

00:40 So if we look at each of these things and i want to inspect the shape, what you need to do is look at kind of the right wings and also the relative location of where the median is in the middle of the box.

00:52 So because there's a large right, right whisker on each of these things, each of these plots is a each of these box plots that's skewed to the right.

01:15 Because the median is kind of left of the box and the right whisker is.

01:24 Is long.

01:26 What that means is there's some extreme values out there pulling it.

01:30 So the mean would actually be further right than the median.

01:37 There's part a in the spread.

01:39 Let's look at this for part b.

01:41 Just keep my same font.

01:42 I'm going to copy and paste this and overwrite it.

01:45 So for part b, we don't want shape, but part b wants us to talk about which one's most likely to have an outlier.

01:52 Okay.

01:55 It looks like one that most likely to have an outlier is going to be the the bmw 3 series.

02:03 So i'm just going to take a second and take that there.

02:05 So this is the bmw 3 series here.

02:12 And because of this super long spread out there in the relative position, so the bmw 3 series is most likely to have an outlier.

02:35 Okay, because of that relatively long whisker.

02:38 And question c talks about young people.

02:42 So young people most likely to buy which car well let's look at the relative position of this so keep my updating here so this is the bmw 5 series and just to compare contrast as i'm labeling this well we're looking at the relative ages here so the relative age we see that there's the median age for the bmw 7 series so that's actually the oldest and this question is about the youngest well the median age is a representation someone in their 40s but if you look at our bmw 3 it looks like someone in their late 20s will most likely have the bmw 3.

03:23 Obviously their spread of all ages, but the central value is the decision maker for that.

03:30 So the bmw 3 series is younger people, because the left median is the most left of those.

04:00 And older people like the other cars.

04:02 There's kind of a tie between those other two.

04:05 And then question d wants us to look at the bmw 5 series.

04:07 So the middle one, which quarter has the smallest spread of data? well, let's look at that.

04:12 So the smallest spread of data.

04:16 Look at this right here.

04:17 So the smallest spread of data, i'm just going to put a little circle around it right there for question d.

04:23 So this small spread, that's first quarter and second quarter.

04:26 So for question d, i'm going to answer second quartile of the small spread.

04:52 I think it's going to keep us looking at that second graph of the bmw 5.

04:58 No one else says what is the spread of that data? so it looks like if we had to estimate the spread of the second quartile data, this is about three years.

05:07 So i'm going to kind of hover over this.

05:09 So three years, i think i looked at it before, is 41 to 38.

05:12 Again, your book in the book shows a little cleaner scale there, but i estimate that to be about three years.

05:27 All right, so then let's keep going here.

05:30 I believe question e is still talking about bmw5...

Please give Ace some feedback