Analyzing a cumulative relative frequency graph

- 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.