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# Estimating mean and median in data displays

AP.STATS:
UNC‑1 (EU)
,
UNC‑1.M (LO)
,
UNC‑1.M.2 (EK)
CCSS.Math:

## Video transcript

we are told researchers scored 31 athletes on an agility test here are their scores it's in this histogram and what I'm going to ask you is which of these intervals interval a B or C which one contains the median of the scores and which one or give an estimate of which one contains the mean of the scores pause this video and see if you can figure that out so let's just start with the median remember the median you could view as the middle number or if you have an even number of data points it would be the average of the middle two here we have an odd number of data points so it would be the middle number so what would be the middle number if you were to order them from least to greatest well it would be the one that has 15 on either side so it would be the sixteenth it would be the sixteenth data point sixteenth data point and so we could just think about which interval here contains the sixteenth data point you could view it for the sixteenth from the highest or the sixteenth from the lowest it is the middle one all right so let's start from the highest so this interval C contains the 13 highest data points and then interval B goes from the 14th highest all the way to the 18th highest so this B contains the median it contains the 16th highest data point or if you started from the left it would also be the 16th lowest data point so that's where the median is the median now what about an estimate for the mean well you have calculated the mean in the past but when you're looking at a distribution like this when you're looking at a histogram one way to think about the mean is it would be the balancing point if you imagine that this histogram was made out of some material of let's say uniform density where would you put a fulcrum in order to balance it if you put the fulcrum right over here it feels like you would have it feels like you would tip over to the left because this is a left skewed distribution you have this long tail to the left if you really wanted to balance it out it seems like you would have to move your fulcrum in the direction of that left skew in the direction of the tail and so I would estimate to balance it out it would actually be closer to that which would be interval a interval a would contain the mean the intention of this type of exercise isn't for you to try to calculate every data point in fact they don't give you all the information here and add them all up and then divide by 31 it's really to estimate and to also get the intuition that when you have a left skewed distribution like this you will often see a situation where your mean is to the left of the median if you have a right skewed distribution it would be the other way around and as we will see when you see a symmetric distribution the mean and the median will be awfully close to each other or when you have a roughly symmetric distribution if you have a perfectly symmetric distribution they might be exactly in the same place so let's do another example so here it says we have the ages of 14 co-workers and what I want you to do is say roughly where is the mean and roughly where is the median is it roughly at a is it roughly at B or is it roughly at C pause this video and try to figure it out so let's first start off with the median we have 14 data points so this would be the average of the middle two data points it was the it would be the average of the seventh and eighth data point well you could say one two three four five six seven and then the eighth one is here so the seventh data point is a 30 the eighth one is in the thirty one bucket so the average of the two would you get U to be another way that you could think about it is you can just eyeball it and see you have just as many data points below B as you do have above B and so that also gives you a good indication that B would be where the median is so that is where the meeting is now what about the mean well this is a perfectly symmetric distribution if I wanted to balance it I would put the fulcrum right in the middle so I would say that the mean would also the mean would also be at B and we are done
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