# Extended sample; mode, median, mean

Given a sample (the scope is 72 elements) with mode=54 mean=55,7 median=54,5. The 73th value of the extended sample is 56. What can I say about the mode, median and mean of the extended sample?

Well, the updated mean is easy to calculate.

To make a statement about the median I know the 36th and the 37th value, since the mean of those two values gives me the median, because I have an even number of observations. I would get the equation (x+y)/2=54,5 which doesnt help me in working out the 37th value from what I see.

There have to be at least two values of 54, so that it could be the mode of the sample (assuming all other values appear just one time in the sample).

• Use definitions of mode, median and mean. Do you e.g. know how the mode is defined? – MerylStreep Oct 17 '15 at 16:26
• The mode is the value (in this case 54) that appears most often in my sample. – craaaft Oct 17 '15 at 16:52

The mean of $n = 72$ observations is $$\bar X_{72} = \frac{\sum_{i=1}^{72} X_i}{72} = 55.7.$$ You can use this information to find $\sum_{i=1}^{72} X_i$ and from there to find $\sum_{i=1}^{73} X_i$ and from there $\bar X_{73}.$

To find the median of all 73 observations you need to know observation number $74/2 = 37$ of the sorted data. I think you may be expected to assume all of the $X_i$s are integers. What does the given information tell you about the middle two observations when the original 36 are sorted?

Finally, what is the minimum number of the original observations that must have had value 54?

In general, problems that ask you to 'update' the mean (and standard deviation) of a sample upon including one more observation are always solvable. But for the median and the mode you cannot always update; you have to rely on quirks of the particular sample in question to see if updating is possible. Do you have enough useful quirks here?

• You might show the numerrical result in your Question along with any other thoughts to show some engagement. And when you have what you need, please click the check-mark to Accept your favorite Answer, so we will take this out of the queue of questions that need more attention. – BruceET Oct 18 '15 at 1:09
• Well, the updated mean is easy to calculate. To make a statement about the median I know the 36th and the 37th value, since the mean of those two values gives me the median, because I have an even number of observations. I would get the equation (x+y)/2=54,5 which doesnt help me in working out the 37th value. There have to be at least two values of 54, so that it could be the mode of the sample (assuming all other values appear just one time in the sample). – craaaft Oct 18 '15 at 1:14
• I'm sorry I just edited the answer to early before writing up all my thoughts. – craaaft Oct 18 '15 at 1:15
• I dont think there are enough quirks to make a statement about the new mode and median. Is this correct? – craaaft Oct 18 '15 at 17:00
• Some info: Median is 56.5 and there's at least two 54's. Using parens in subscripts to denote rank order, we know $X_{(36)} = 54, X_{(37)} = 56$ in orig sample of 72. For the new sample of 73, we have another value at 56. You can take it from there. – BruceET Oct 18 '15 at 19:22