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## Statistics and probability

# Analyzing a cumulative relative frequency graph

Find percentiles, median, quartiles, and IQR using a cumulative relative frequency graph.

## Want to join the conversation?

- How did you find the first and third quartile?(8 votes)
- when we look at a data set, we split it into what we call a five number summary: the minimum, the first quartile, the median, third quartile, and maximum.

by looking at it this way, we can see that the first and third quartiles are the values that are directly between the median and the minimum/ maximum (in other words, what is exactly at 25% and 75%).

So, on the graph, we simply find 25% and 75% and write those values, in this case, 18 and 39.(16 votes)

- Where does the name "Cumulative relative frequency" come from, how can it be interpreted?.

I mean what is a good way to think about it in order to remember what is it about?...(11 votes)- the point say, 50 grams of sugar, will have a y-axis of the number of drinks with at most 50 grams of sugar in them. Though I'm not sure where the relative part comes from(2 votes)

- What does it mean by "cumulative relative frequency"?(3 votes)
- Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.(9 votes)

- where did you get 25% and 75% from?(5 votes)
- 25% is the first quartile and 75% is the third. You find the IQR by subtracting the first from the third quartile.(4 votes)

- I have a question about a Excel thing we have to draw.. and it is awfully confusing. I am trying to draw a frequency chart. and then a histogram. I can't seem to get it right. The numbers are (a) Construct a frequency distribution for the number of different residences occupied by graduating seniors during their college career:

1, 4, 2, 3, 3, 1, 6, 7, 4, 3, 3, 9, 2, 4, 2, 2, 3, 2, 3, 4, 4, 2, 3, 3, 5.(4 votes) - How do we fill in a table of cumulative frequency?(4 votes)
- For each unique value in your data, count how many times it appears (frequency), then divide it by total frequency to get relative frequency.

Start filling the table with the frequency of the first value. For the next, add the frequency of the next value to the previous row's cumulative frequency. Repeat this for all values.(1 vote)

- How do you find the first and third quartile? Plz help.(2 votes)
- Find the median then use the median to split the data into two equal parts with the median separating them.

Now for each of these two parts find the median again. The smaller one is the first quartile and the larger is the third quartile.

Does that make sense? Or would you like a demonstration?(5 votes)

- I don't get Cumulative frequency because in my book it says The Cumulative Freq is the sum of all added up to and including the current one(2 votes)
- Yes, the cumulative frequency of 20 grams of sugar is equal to the number of drinks that contain 20 grams of sugar or less.

To get the cumulative*relative*frequency of 20 grams of sugar, we divide that number by the total number of drinks, namely 32.

From the graph, we see that the cumulative relative frequency of 20 grams of sugar is approximately 0.3, which means that about 30% of the 32 drinks contained at most 20 grams of sugar.

30% of 32 = 9.6, so 9 or 10 of the drinks contained 20 grams of sugar or less, which is the cumulative frequency of 20 grams of sugar.(5 votes)

- How do you plot a cumulative frequency graph from a table of values that contains cumulative frequency,(3 votes)
- In class, we learned about a cumulative-distribution function. Is this the same as cumulative relative frequency?(2 votes)
- No, a cumulative-distribution function is a function whose value is the probability that a corresponding continuous random variable has a value less than or equal to the argument of the function.(2 votes)

## Video transcript

- 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.