Is it possible to calculate Q1, Median, Q3, StDev from already aggregated data?

We have data that will get aggregated per hour into the following values

• Q1
• Median
• Mean
• Q3
• Standard Deviation
• Max
• Min
• Count of Values

So the data will look more or like this in the end.

        00:00-01:00     01:00-02:00     02:00-03:00     03:00-04:00          ...
--------------------------------------------------------------------------------
Q1            68,72           69,64           64,31           64,40          ...
Median       118,72          124,42          115,54          118,11          ...
Mean         119,17          119,97          117,23          117,60          ...
Q3           169,64          171,72          170,63          168,72          ...
StDev         59,30           59,15           61,23           59,62          ...
Max          219,70          219,44          219,76          219,71          ...
Min           15,02           15,07           15,05           15,05          ...
Count       1000,00         1000,00         1000,00         1000,00          ...


Now we want to aggregate the same values for a whole day (24h) without using the original data if possible (because in our real scenario it would require a significantly longer time to aggregate from those).

For most of them it's pretty straight forward, like MIN is simply the overall MIN, AVG is the overall AVG, etc.

But the tricky part is Q1, Median, Q3 and StDev.

From what I understand it's not possible to simply calculate the (weighted) average value of the 24 separate values. But is there a method to achieve this from already aggregated values (for example by storing some additional data)?

Is the difference from such a huge dataset even significant?

Or will the data always be distorted except for calculating it from the whole dataset?

You have mean, counts and StDev of the observations, so aggregated StDev is a matter of algebra. I'm sure you can figure it out easily.

The quantiles are trickier. Consider, Q1 of two samples. They form the bounds of the Q1 of the combined sample. If $Q1_1>Q1_2$, then it's easy to see that aggregated $Q1_2<Q1$ and $Q1<Q1_1$. That's all you can say about the quantiles, i.e. in your case $min(Q1_i)<Q1<max(Q1_i)$.

You can get a little more from your data by using asymptotic sample quantile distribution. In this case instead of getting the bounds, you could estimate the StDev of the quantiles. You'd have to assume that the distribution doesn't change during the day.

Alternatively, you could try to estimate the quantiles during the day, e.g. they're higher in the morning and lower in the evening. In this case, you could run a test to see whether this is the case.

• The Standard deviation (or at least the variance, from which you can work it out) has been covered in a number of questions already on site, it may be worth linking to one. – Glen_b May 12 '15 at 14:13