The figure below is a cumulative relative frequency graph of...

The figure below is a cumulative relative frequency graph of the amount spent by 50 consecutive grocery shoppers at a store. (a) Estimate the interquartile range of this distribution. Show your method. (b) What is the percentile for the shopper who spent $19.50? (c) Challenge: Draw the histogram that corresponds to this graph.

The figure below is a cumulative relative frequency graph of the amount spent by 50 consecutive grocery shoppers at a store.
(a) Estimate the interquartile range of this distribution. Show your method.
(b) What is the percentile for the shopper who spent $19.50?
(c) Challenge: Draw the histogram that corresponds to this graph.

Key Concepts

Cumulative Relative Frequency Graph

A cumulative relative frequency graph is a tool used in statistics to display the accumulation of data points. This graph plots the cumulative percentage of observations up to each data value, helping to visualize how data accumulates over a range of values and to easily determine percentiles or quartiles from the distribution.

Interquartile Range

The interquartile range (IQR) is a measure of variability and is defined as the difference between the third quartile (75th percentile) and the first quartile (25th percentile). It represents the middle 50% of the data and is less affected by outliers, making it a robust indicator of the spread of a distribution.

Percentile

A percentile indicates the relative standing of a value within a data set. It shows the percentage of data points that fall below a specific value. For instance, if a value is at the 40th percentile, then 40% of the observations in the dataset are below that value, providing insight into the position of that data point within the overall distribution.

Histogram

A histogram is a graphical representation of the distribution of numerical data. It is created by dividing the range of data into intervals (bins) and counting how many data points fall into each bin. Histograms provide a clear visual understanding of data distribution, including its shape, central tendency, and variability, and are often used to complement cumulative graphs.

Transcript

00:01 All right, so in this problem, we are looking at a relative frequency graph for the amount of money that grocery shoppers spend in a store.

00:08 And the first thing we're going to do is we're going to find the iqr.

00:12 Okay, and remember the iqr is the 75th percentile minus the 25th percentile.

00:20 So then what we need to do on our chart is we need to look at, well, what's the 75th percentile? okay, if we come over here, it's right before 50, i would say about 48.

00:29 All right so $48 and then the 25th percentile is right about 20 and then 48 minus 20 would be 28 so the iqr in this case would be 28 all right so for percentile we are going to be looking at the the amount is 1950 so that's right below 20 and if we go up here you see it's right between 20 and 30 it looks similar to the picture in the book and that's actually the one we used for the first part there in number one which is between 20 and 30 so i said 25th percentile anything in that range plus or minus two or three percent points would would be fine i think and then for part c we are going to create a histogram so it's going to look very similar in terms of the scale so we're still going to go from 10 to 100 on the x axis okay but then on the y axis it's going to be the number of customers as opposed to the percentage of customers so i'm gonna do this actually in red so we look at between zero and ten that's a little bit less than ten percent and there were fifty total so one two three four five six six i think the most we're going to probably have is about 10.

02:17 All right.

02:18 So between zero and 10, that looks like a little bit less than 10%.

02:23 So a little bit less than 5, we'll go about four.

02:27 Okay.

02:28 So between zero and $10, about four customers spent that much.

02:33 And then we increase all the way up to about 30%.

02:39 So that's close to about 20%.

02:42 20 % of 50 is 10.

02:46 This is where we're going to have a little bit more there.

02:53 And then it looks like we increased from about 30 to 60%, which is about 30%...