Standard Deviation Calculations?????

[Yorric]

Hi all
I just got a Chrony & Ballistic Printer & it gives me lots of nice numbers to play with & prints them out just fine.
To get to know it I just ran some air rifle shots over it & it works fine. Ten shots with print outs of each, then min, max, average, extreme spread & standard deviation (all to two decimal figures).
Then just to see if the little grey cells work well enough, I put the shot velocity data into an Excel spreadsheet & worked out the Std Dev there. I also used my scientific calculator to determine SD on there.
The three ways of calculating all gave different results!!
Chrony says SD = 6.55
Excell says SD = 6.478369908
Casio says SD = 6.14592133
wtfwtf
I expect the differences must be down to the way that the three systems round up or down the input data values.
What is surprising is the extent of the differences.

Any comments from you boffins out there?

The old saying "There'se lies, damn lies & statistics!"

Ian

[zanes]

Chrony I could understand perhaps (not sure what "oomph" they have inside but it wouldn't surprise me if they were working to a low precision for the "derived" figures- in any case only .08 out from excel so a fair estimation), but I'm surprised by the difference between excel and your calc (assuming it's a decent casio and so working to a fair number of decimal places internally). You haven't messed up any operator priorities (BODMAS!)?

Statistics are really not my bag though....

Got the raw figures?

[Yorric]

Raw figures as printed on Chrony
406.20
400.92
401.41
404.81
409.54
385.88
402.88
401.82
395.54
401.69

I suppose I could be doing summat wrong with the Casio scientific calculator - It's way too sophisticated for me so I very rarely use it. But it has a specific instruction for SD so it should have worked????

[FencepostError]

Wolfram Alpha seems to agree with Excel (within given digits)

http://www.wolframalpha.com/input/?i=st ... %2C+401.69

(Sure there's no data entry problem using the calculator?)

Edit: Ah! Got it! Your calculator is doing population standard deviation instead of sample standard deviation! Wolfram Alpha will replicate the calc's result if you specify click on the "use as population standard deviation" link.

(It seems you'll want sample standard deviation in this case.)

[Yorric]

Many thanks for your replies. It looks like I need to forget playing with the calculator then! - As I said before it certainly falls within the "too difficult box" for a duffer like me. HNC maths/statistics of many moons ago didn't teach different types of Standard Deviation systems!!
For all practical purposes I'll just believe the Chrony for my records as it agrees with Excel (nearly).

Ian

[Laurie]

Yes, but mathematical curiosity aside, what practical differences do the three SD values actually make on a (very modest) sample of 10 values. SD in itself is meaningless until applied as a qualifier to the arithmetic mean. The SD value 'predicts' (good word!) that 68.26% of a sample's population will fall within the plus or minus SD value of the mean, in this case 401.07.

So a bit over two thirds of the next 10 shots are expected to fall within a 12.3 fps range around the average, or at the other extreme of the three SD values, they'll fall within a 13.1 fps range. The other nearly a third of shots should fall outside of that. Wow!

The prediction only works if the total sample values produce a bell shaped curve if graphed. Apart from a sample of 10 being far too small to see what sort of curve a range of MVs produce, looking at the series immediately shows it's not a bell shape as there is a low velocity shot that is way out of the rest of the pattern, 385.88, and there is a high shot at 409 that doesn't fit very well either.

That assumes one believes the chronograph, of course. I have doubts about the consistency of many makes of optical sensor types with somewhat larger centrefire cartridge bullets never mind airgun pellets.

[billgatese30]

The data you have collected is normally distributed (standard bell curve that Laurie refers too) but only just, your 388.85 reading is skewing the distribution due to the low sample size (10 data values). However it does mean that statistically the data is normal so that the sample standard deviation applies so as Laurie also pointed out, 68% of your shots should produce between 394.6-407.5 fps, 95% of your shots would be within 388.1-414.0 fps and 99.7% of your shots would be within 381.6-420.5 fps. This means that the vast majority of your rounds fired under these exact conditions, you should be able to expect 997 out of 1000 to have anywhere up to 38.87 fps difference to any other random round from that batch.

However I should point out that a larger sample size would give a more accurate prediction, also, a different sample of the same population would also give a different prediction of the same population. Although with a large enough sample, hopefully different samples from the same population should give predictions as close to each other as to make little significant difference in real world figures.

[meles meles]

Maybe we should ask Mystic Meg, or that octopod?

[Yorric]

Hi again boffinpeeps,
I am well aware of the need for adequate sample sizes for any useful statistical analysis. My only concern in my original post was the difference in the calculated results by the three systems. That has been resolved very clearly by FencepostError - thanks.
The string of figures I used were what was measured - using a very cheap air rifle just to get used to the workings of the printer - hence the relatively "poor" results - and the chrony read the velocities reliably with no hiccups.
I now know the way forward for me is to believe the calculations of my Chrony and if I need more I can also trust Excel.
Having used a non-printing Chrony for many years I know I can trust the system.
I do intend to run some tests shooting bullets over two Chronys in series as a test to see if they both read the same with the same bullet.
I see these instruments as comparators primarily with the added bonus of some actual measurement output - I do not expect fully calibrated measurements.

Ian