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PostPosted: Mon Nov 18, 2013 12:37 pm 
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Does anyone know of the highest ever mean Standardised Score a DD has ever achieved for Kendrick?

I'm just curious :)


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PostPosted: Sat Nov 30, 2013 5:47 pm 
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sonasona wrote:
Does anyone know of the highest ever mean Standardised Score a DD has ever achieved for Kendrick?

I'm just curious :)


I can give you a statistically probable answer which should be very close. Standardised scores are worked out by working out the standard deviation which is basically the average amount by which the scores deviate from the mean. There's a good description of what standard deviation is here:
http://www.mathsisfun.com/data/standard-deviation.html
In simple terms the bigger the standard deviation the more spread out the underlying raw scores would have been.

In just about every natural distribution; the width of stripes on zebras, diameter of smarties or the speed of cars on the M4 you invariably end up with the familiar bell curve. Exactly the same is true for entrance test results. There are very predictable probabilities of the numbers in any part of the curve and in an sample of say 700 you can pretty much guarantee that 68% of the scores will be within ± (plus or minus) one standard deviation, ie between 85 and 115. It's not quite so predictable at the thin tails of the top and bottom of the bell curve but statistical probability predicts that 0.62% of the cohort (4.34 girls!) will score at or below -2.5 standard deviations and 4.34 girls will score at or above +2.5 standard deviations. Obviously you can't have 0.34 of a girl (unless you've read the Phantom Tollbooth) but I'd be very confident in saying that the range of scores is ± 2.5 standard deviations. This is usually a bit meaningless to most people so by convention test results are usually scaled so that that the mean is 100 and each standard deviation is 15, (sometimes referred to as the Wechsler scale.) So having concluded that that given a cohort of 700 the range of scores is going to be be ± 2.5 standard deviations the actual range of standardised scores is then 100 ± 2.5 × 15 or from 62.5 lowest to 137.5 highest.

Eventually ;-) the answer to your question should therefore be 137.5.

Put a less mathematical way imagine you plot all the scores on a graph but leave the scales out. When you come to put the scale in -2.5 standard deviations (or 62.5) needs to go at the bottom of the scale and +2.5 (or 137.5) has to go at the top. If for some bizarre reason one year the girls correctly answered twice as many questions the lowest score would still be 62.5 and the highest would still be 137.5. The scores will (probably) always be in this exact range unless there is something weird going on. ("Weird" could be a number of different things but to give an example if all the questions were so easy that half the girls answered every single one correctly you'd end up with a bell curve that looked like someone had run over it with a lawnmower. NFER set the Kendrick tests and they are a professional and competent (unlike the local boy's school) so I'd be very confident there is nothing weird.

Disclaimer!
Whilst in all probability if I throw a dice six times I'll manage get a six that isn't guaranteed!

Having posted this I'd really be interested in anyone confirming or trashing the theory but getting someone to post an answer is entirely reliant upon one of the proud parents of the 4.34 who's girls happened to have the highest scores in any year reading this and posting a reply. (Mine's only seven but early indications are good ;-0) What are the odd of that? (... actually 4.34/700 × the proportion of applicants who read this post ... OK pretty slim I think!)

If you *really* want to know the actual answer you will probably need to ask the school. I can't see any reason for them not wanting to tell you even if your motive is just curiosity so you probably don't need to mention the FoI Act in order to get them to answer this.

Hope that wasn't too mathematical !


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PostPosted: Sat Nov 30, 2013 11:23 pm 
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That's brill jamesberks!

Let's see how close we can get to this number next year then :wink:


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PostPosted: Sun Dec 01, 2013 10:23 am 
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Location: Reading
@Jamesberks, I'm no stats expert. Loved the rest of maths but not stats.

Does your reasoning deal with the fact that the Kendrick score is an average of two tests rather than a single test score.

For someone to get 141 as an average score they would need to get 141 in both tests.


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PostPosted: Sun Dec 01, 2013 1:54 pm 
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I haven't looked at Kendrick's processes in detail and so the above was more theoretical. A quick glance at the first five pages of viewtopic.php?f=10&t=34518 shows that every single final score is the simple arithmetic mean of the other two. They could equally well have just added the two tests and arrive at the same ranking although with a mean of 200 and standard deviation of 30. The Wechsler scale is just convenient as it's familiar because that's what everyone uses (apart from Reading School who are very creative in that respect.)

As you correctly point out in order to get the maximum possible final score a girl has to score the maximum in both tests. How may correctly answered questions that equates to depends on how the whole group performed on the day but if you know the sample size, mean and standard deviation you can quickly calculate the theoretical distribution using Excel's NORMDIST() function. For example using some very plausible values such as a cohort of 700, mean of 30 and standard deviation of 5 you'd end up with this distribution:

(number of marks => number of girls with that score)

10 => 0.02
11 => 0.04
12 => 0.09
13 => 0.17
14 => 0.33
15 => 0.62
16 => 1.11
17 => 1.9
18 => 3.14
19 => 4.97
20 => 7.56
21 => 11.05
22 => 15.53
23 => 20.96
24 => 27.19
25 => 33.88
26 => 40.56
27 => 46.65
28 => 51.56
29 => 54.75
30 => 55.85
31 => 54.75
32 => 51.56
33 => 46.65
34 => 40.56
35 => 33.88
36 => 27.19
37 => 20.96
38 => 15.53
39 => 11.05
40 => 7.56
41 => 4.97
42 => 3.14
43 => 1.9
44 => 1.11
45 => 0.62
46 => 0.33
47 => 0.17
48 => 0.09
49 => 0.04

The model works really well in the middle where you have 50+ getting that number of answers for each question but is a bit more random towards the ends. Adding some common sense interpretation the lowest score would be around 14 or 15 and the highest 45 or 46.

I didn't know anything about standardised scores before May this year but found myself with a sudden interest in the subject. ;-)


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PostPosted: Sun Dec 01, 2013 5:28 pm 
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Surely the highest raw score is of more interest?


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PostPosted: Sun Dec 01, 2013 9:44 pm 
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Guest55 wrote:
Surely the highest raw score is of more interest?


Yes, but you'd have to ask the school that. ;-)


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