Analyzing a cumulative relative frequency graph | AP Statistics | Khan Academy
Summary
TLDRThe video script discusses a cumulative relative frequency graph for sugar content in 32 Starbucks drinks. It explains how to read the graph to determine the percentile of a drink with 15 grams of sugar, identifying it as the 20th percentile. The script also estimates the median sugar content to be approximately 25 grams, as 50% of drinks have 25 grams or less. Lastly, it calculates the interquartile range, estimating the 25th percentile at around 18 grams and the 75th at 39 grams, resulting in an interquartile range of about 21 grams.
Takeaways
- π The script discusses a cumulative relative frequency graph for sugar content in 32 Starbucks drinks.
- π¬ It explains that 0% of the drinks have no sugar content, and 10% have 5 grams or less.
- π― The percentile of an iced coffee with 15 grams of sugar is estimated to be in the 20th percentile based on the graph.
- π The median sugar content, representing the middle value of the distribution, is approximated to be 25 grams.
- π The interquartile range is calculated by finding the 25th and 75th percentiles, which are estimated to be around 18 grams and 39 grams, respectively.
- βοΈ The estimated interquartile range, the difference between the first and third quartiles, is about 21 grams.
- π The script provides a step-by-step guide on how to interpret a cumulative relative frequency graph to find percentiles and the interquartile range.
- π Understanding the graph involves recognizing that each point represents the percentage of drinks with that amount of sugar or less.
- π The cumulative nature of the graph means that as sugar content increases, the relative frequency also increases, showing the proportion of drinks with that sugar level or lower.
- π The script uses the example of an iced coffee to demonstrate how to estimate the percentile for a specific sugar content.
Q & A
What is the purpose of the cumulative relative frequency graph discussed in the script?
-The cumulative relative frequency graph is used to show the distribution of sugar content in grams for 32 Starbucks drinks, illustrating how many drinks contain a certain amount of sugar or less.
How is the percentile of an iced coffee with 15 grams of sugar estimated in the script?
-The percentile is estimated by finding the cumulative relative frequency that corresponds to 15 grams of sugar on the graph, which is approximately 20%, placing the iced coffee in the 20th percentile.
What does the 50th percentile represent in the context of the Starbucks drinks data?
-The 50th percentile represents the median sugar content, where half of the drinks have 25 grams or less of sugar, based on the cumulative relative frequency graph.
How is the interquartile range of the distribution of drinks estimated from the script?
-The interquartile range is estimated by identifying the 25th and 75th percentiles on the graph, which are approximately 18 grams and 39 grams, respectively. The difference between these two values gives an estimate of the interquartile range, which is about 21 grams.
What does the term 'cumulative relative frequency' mean in the context of the script?
-In the context of the script, 'cumulative relative frequency' refers to the proportion of drinks that have a certain amount of sugar or less, as represented on the y-axis of the graph.
What is the significance of the 0.5 on the vertical axis of the cumulative relative frequency graph?
-The 0.5 on the vertical axis signifies the 50th percentile, indicating that 50% of the drinks have a sugar content of that amount or less.
How does the script describe the process of estimating the median from the cumulative relative frequency graph?
-The script describes estimating the median by locating the point on the graph where the cumulative relative frequency is 0.5 (50%), which corresponds to the sugar content that half of the drinks have or less.
What is the estimated sugar content for the 25th percentile of Starbucks drinks according to the script?
-The script estimates the sugar content for the 25th percentile to be approximately 18 grams of sugar.
What is the estimated sugar content for the 75th percentile of Starbucks drinks according to the script?
-The script estimates the sugar content for the 75th percentile to be approximately 39 grams of sugar.
How does the script suggest interpreting the cumulative relative frequency graph for understanding the distribution of sugar content?
-The script suggests interpreting the graph by converting the cumulative relative frequencies to percentages and understanding that these percentages represent the proportion of drinks with a certain amount of sugar or less.
Outlines
π Understanding Cumulative Relative Frequency Graphs
This paragraph discusses the interpretation of a cumulative relative frequency graph for sugar content in 32 Starbucks drinks. The graph shows the percentage of drinks with a certain amount of sugar or less. For instance, 10% of drinks have 5 grams or less of sugar. The percentile of an iced coffee with 15 grams of sugar is estimated to be the 20th percentile, as 20% of drinks have 15 grams or less. The median is determined by looking at the 50th percentile, which is approximately 25 grams, meaning half of the drinks have 25 grams or less of sugar.
π Estimating the Interquartile Range from Cumulative Data
The second paragraph focuses on estimating the interquartile range from the cumulative relative frequency graph. The interquartile range is calculated by finding the 25th and 75th percentiles and then taking the difference between them. The 25th percentile is estimated to be around 18 grams, with 25% of drinks having 18 grams or less of sugar. The 75th percentile is estimated to be around 39 grams, with 75% of drinks having 39 grams or less. The difference between these two points, which is the interquartile range, is approximately 21 grams. The best estimate from the given choices for the interquartile range is 20 grams.
Mindmap
Keywords
π‘Nutritionists
π‘Sugar Content
π‘Cumulative Relative Frequency Graph
π‘Percentile
π‘Median
π‘Interquartile Range
π‘Data Point
π‘Estimation
π‘Quartiles
π‘Contextual Understanding
Highlights
Nutritionists measured the sugar content in 32 Starbucks drinks.
A cumulative relative frequency graph is used to represent the data.
Zero percent of drinks have no sugar content.
10 percent of drinks have 5 grams or less of sugar.
100 percent of drinks have 50 grams or less of sugar.
Cumulative relative frequency shows the percentage of drinks with a certain sugar amount or less.
An iced coffee with 15 grams of sugar is in the 20th percentile.
The median sugar content is estimated to be 25 grams, representing the 50th percentile.
The 25th percentile is approximately 18 grams of sugar.
The 75th percentile is roughly 39 grams of sugar.
The interquartile range is estimated to be 21 grams.
The cumulative relative frequency graph helps in estimating percentiles and the median.
The graph shows a gradual increase in relative frequency as sugar content increases.
The median is determined by the point where 50 percent of drinks have less or equal sugar content.
The interquartile range is calculated by finding the difference between the 25th and 75th percentiles.
The estimated interquartile range of 20 grams is the best match based on the graph.
The data provides insights into the distribution of sugar content in Starbucks drinks.
Transcripts
- Nutritionists measured the sugar content
in grams for 32 drinks at Starbucks.
A cumulative relative frequency graph, let me
underline that, a cumulative relative frequency graph
for the data is shown below.
So, they have different on the horizontal axis,
different amounts of sugar in grams and then,
we have the cumulative relative frequencies.
Let's just make sure we understand how to read this.
This is saying that zero or zero percent of the drinks
have a sugar content, have no sugar content.
This right over here, this data point,
this looks like it's at the .5 grams
and then this looks like it's at 0.1.
This says that 0.1, or I guess we could say
10 percent of the drinks that Starbucks
offers has five grams of sugar or less.
This data point tells us that a hundred
percent of drinks at Starbucks has 50 grams
of sugar or less.
The cumulative relative frequency, that's why
at each of these points we say this is the frequency
that has that much sugar or less.
And, that's why it just keeps on increasing
and increasing as we add more sugar
we're going to see a larger portion
or a larger relative frequency has that
much sugar or less.
So, let's read the first question.
An iced coffee has 15 grams of sugar.
Estimate the percentile of this drink
to the nearest whole percent.
So, iced coffee has 15 grams of sugar
which would be right over here.
And so, let's estimate the percentile.
So, we can see they actually have
a data point right over here and
we can see that 20 percent or 0.2,
20 percent of the drinks that Starbucks
offers has 15 grams of sugar or less.
So, the percentile of this drink,
if I were to estimate it, looks like
it's the relative frequency 0.2 has
that much sugar or less so this percentile
would be 20 percent.
Once again, another way to think about it,
to read this you could convert these to percentages.
You could say that 20 percent has this much sugar or less.
15 grams of sugar or less, so an iced coffee
is in the 20th percentile.
Let's do another question.
So here, we are asked to estimate the median
of the distribution of drinks.
Hint to think about the 50th percentile.
So, the median, if you were to line up all
of the drinks, you would take the middle drink.
And so, you could view that as well, what
drink is exactly at the 50th percentile?
So, now let's look at the 50th percentile
would be a cumulative relative frequency of 0.5,
which would be right over here on our vertical axis.
Another way to think about it is 0.5, or 50 percent
of the drinks are going, if we go to this point
right over here, what has a cumulative relative
frequency of 0.5.
We see that we are right at looks like this is 25 grams.
So, one way to interpret this is 50 percent
of the drinks have less than or have 25
grams of sugar or less.
So, this looks like a pretty good estimate
for the median, for the middle data point.
So, the median is approximately 25 grams
that half of the drinks have 25 grams or less of sugar.
Let's do one more based on the same data set.
So, here we're asked, what is the best estimate
for the interquartile range of the distribution of drinks?
So, the interquartile range, we wanna figure out
well, what's sitting at the 25th percentile?
And we wanna think about what's
at the 75th percentile, and then we
want to take the difference.
That's what the interquartile range is.
So, let's do that.
So, first the 25th percentile, we wanna look
at the cumulative relative frequency, so 25th
this would be 30th, so the 25th would be right
around here and so, it looks like the 25th
percentile is that looks like about, I don't know,
and we're estimating here, so that looks like it's
about this would be 15, I would say maybe 18 grams.
So, approximately 18 grams.
Once again, one way to think about it is,
25 percent of the drinks have 18 grams of sugar or less.
Now, let's look at the 75th percentile.
So it's the 70th, 75th would be right over there.
Actually, I can draw a straighter line than that.
I have a line tool here.
75th percentile would put me right over there.
I don't know, that looks like, I'll go with
39 grams, roughly 39 grams.
And so, what's the difference between these two?
Well, the difference between these two, it looks
like it's about 21 grams.
So, our interquartile range, our estimate
of our interquartile range, looking at this
cumulative relative frequency distribution,
'cause we're sayin', hey look, it looks like
the 25th percentile, looks like 25 percent
of the drinks have 18 grams or less.
75 percent of the drinks have 39 grams or less.
If we take the difference between these two
quartiles, this is the first quartile,
this is our third quartile.
We're gonna get 21 grams.
Now, if we look at this choice, the choices
right over here, 20 grams definitely
seems like the best estimate,
closest to what we were able to estimate
based on looking at this cumulative
relative frequency graph.
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