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"Half the schools are below average" - not always true
Posted on 9/30/14 at 9:56 am
Posted on 9/30/14 at 9:56 am
NGT's so called lies aside - the statement "half the schools are below average" is NOT automatically true, as the late great George Carlin has implied (though he referred to people in general not schools) and now, I'm guessing, NGT (though I haven't seen the speech, correct me if I'm wrong).
Its pretty easy to see how it can be false.
Here's an example: 4 schools, each with the same population. Three have an average test scores of 90 - while the 4th has an average test score of 70. The average is thus (3*90+70)/4 = 85. Thus only 25% of the schools in this case, are below average.
Its pretty easy to see how it can be false.
Here's an example: 4 schools, each with the same population. Three have an average test scores of 90 - while the 4th has an average test score of 70. The average is thus (3*90+70)/4 = 85. Thus only 25% of the schools in this case, are below average.
This post was edited on 9/30/14 at 1:55 pm
8
Posted on 9/30/14 at 9:58 am to SpidermanTUba
quote:
I'm guessing, NGT
Actually, your OP shows that Tyson doesn't understand averages. The response he's going for in his speeches is "Of course half are below average, duh!".
Posted on 9/30/14 at 10:04 am to TN Bhoy
quote:
Actually, your OP shows that Tyson doesn't understand averages. The response he's going for in his speeches is "Of course half are below average, duh!".
If that is indeed true - then you are correct.
In fact statistics may be the most widely misunderstood concept amongst the educated professions (scientists, doctors, etc).
Nassim Taleb makes a point of this in his book "Fooled by Randomness".
Here's a good example he gives for how doctors don't understand statistics:
Given
A) that the false positive rate for a given test for a disease is 1% - and -
B) that the known rate of that disease in the population is 0.1%
EDIT - C) there are NO false negatives
What are the chances that a patient who tests positively for the disease actually has it?
According to Taleb (I forget if he cites a source or not) - doctors will give the wrong answer almost every time.
This post was edited on 9/30/14 at 10:12 am
Posted on 9/30/14 at 10:07 am to SpidermanTUba
quote:
According to Taleb (I forget if he cites a source or not) - doctors will give the wrong answer almost every time.
Wouldn't the answer be 99%?
Posted on 9/30/14 at 10:08 am to SpidermanTUba
quote:
Its pretty easy to see how it can be false.
yeah, because with a skewed distribution the mean is NOT the central tendency.
However, it has been shown that in large enough sample sizes, and when averaging the scores of the students (i.e., the mean of means) that we come to be able to use something called the Central Limit Theorem.
Dont worry, your 7005 class will cover this in November.
This post was edited on 9/30/14 at 10:09 am
Posted on 9/30/14 at 10:10 am to kingbob
quote:
Wouldn't the answer be 99%?
Nope.
Its a LOT less. 1 in 11.
Proof:
1% false positive means that out of a population, of say, 1000 - 10 people will test false positive.
But since the known disease rate is 0.1% - only one person out of that population of 1000 will actually have the disease.
So there are 10 people who test false positive for every one person who actually has the disease.
(I forgot to mention there are no false negatives in this example)
This post was edited on 9/30/14 at 10:11 am
Posted on 9/30/14 at 10:15 am to SpidermanTUba
quote:
So there are 10 people who test false positive for every one person who actually has the disease.
(I forgot to mention there are no false negatives in this example)
See I misread the question as giving a false negative rather than a positive. I get it now.
Posted on 9/30/14 at 10:20 am to kingbob
Half of all schools score below the Median score.
FACT.
FACT.
Posted on 9/30/14 at 10:24 am to CptBengal
quote:
However, it has been shown that in large enough sample sizes, and when averaging the scores of the students (i.e., the mean of means) that we come to be able to use something called the Central Limit Theorem.
Dont worry, your 7005 class will cover this in November.
Your understanding of the central limit theorem is flawed.
The central limit theorem means that if we take a large enough population of Presidential polls - the results of those polls will be normally distributed.
It doesn't mean if we sample a large enough population, the result of the poll will always tend towards 50/50.
So in this context - it would mean if we created samples of schools by random selection - the distribution of scores among those samples is normally distributed - it doesn't mean the scores themselves are normally distributed. If that's the case - it means there can be no skewed distributions in nature - which is kinda absurd as they are observed all the time.
This post was edited on 9/30/14 at 10:29 am
Posted on 9/30/14 at 10:31 am to SpidermanTUba
quote:
It doesn't mean if we sample a large enough population, the result of the poll will always tend towards 50/50.
Go reread what I said, clown.
When we are getting testing values for schools, we are averaging the testing of the students in that school.
Then by computing the mean value to determine central tendency, we are also computing a mean.
By computing the mean of the means for a large number of students nested within schools and then calculating the central tendency of that higher population we are doing exactly what you suggest by
taking the average of a population of polls. Each school is a population.
good god man.
Posted on 9/30/14 at 10:39 am to CptBengal
quote:
Go reread what I said, clown.
When we are getting testing values for schools, we are averaging the testing of the students in that school.
Then by computing the mean value to determine central tendency, we are also computing a mean.
By computing the mean of the means for a large number of students nested within schools and then calculating the central tendency of that higher population we are doing exactly what you suggest by
taking the average of a population of polls. Each school is a population.
good god man.
Again, you misunderstand statistics.
The central limit theorem only applies to randomly selected samples. Selecting your samples by school isn't random. Seriously dude, that's kinda obvious.
This post was edited on 9/30/14 at 10:41 am
Posted on 9/30/14 at 10:40 am to SpidermanTUba
The larger the sample then the percentage of schools above or below the average of that same sample approaches 50%. Therefore the headline "Half the schools are below average" becomes more and more stupid. That is his point. Unless you believe that your specific group of schools in your state / region should be superior for some reason, seeing that half are below average shouldn't alarm anyone.
Posted on 9/30/14 at 10:40 am to Sid in Lakeshore
quote:
Half of all schools score below the Median score.
Just something for all to consider. Schools and students are not the same pool. For example, private schools tend to have more above average students than public schools. But private schools tend to have much smaller student bodies. Therefore they are over-represented in the pool of 'all schools' vis-a-vis the pool of 'all students.'
And don't forget. All the children in Lake Wobegon are above average.
Posted on 9/30/14 at 10:41 am to SpidermanTUba
quote:
Again, you misunderstand statistics
quote:
The central limit theorem only applies to randomly selected samples. Selecting your samples by school isn't random. Seriously dude, that's kinda obvious.
These samples are randomly collected. Perhaps you dont understand what randomization actually means in this case.
Posted on 9/30/14 at 10:44 am to mmcgrath
quote:
The larger the sample then the percentage of schools above or below the average of that same sample approaches 50%
NOT GENERALLY TRUE
This only happens if the underlying distribution of scores is normal.
quote:
...seeing that half are below average shouldn't alarm anyone.
I agree. In fact that's a GOOD thing, IMO.
Posted on 9/30/14 at 10:45 am to SpidermanTUba
quote:
NOT GENERALLY TRUE
This only happens if the underlying distribution of scores is normal.
Look up the binomial approximation towards normal and get back to us....
Posted on 9/30/14 at 10:47 am to CptBengal
quote:
These samples are randomly collected.
Uhh, no. In the example you provided, the samples are selected based on school.
quote:
Perhaps you dont understand what randomization actually means in this case.
Apparently it means grouping samples of students based on school - rather than based on random selection.Posted on 9/30/14 at 10:50 am to SpidermanTUba
quote:
Uhh, no. In the example you provided, the samples are selected based on school.
WHat is the sampling unit in my example (it seems we will have to go VERY basic for you) quote:
pparently it means grouping samples of students based on school - rather than based on random selection.
Look up the concept of nesting as it relates to randomization and get back to me champ. BTW, these are intro level stats questions. The fact you have such difficulty with them is astoundign for a PhD in astrophysics.
Posted on 9/30/14 at 10:52 am to CptBengal
quote:
Look up the binomial approximation towards normal and get back to us....
The test scores of students within a school are NOT INDEPENDENT variables. OBVIOUSLY. They have the same teachers, the same learning conditions, the same fellow students to study with (or not). You are populating your samples based on which school a student attends - NOT RANDOM.
Posted on 9/30/14 at 10:56 am to CptBengal
quote:
WHat is the sampling unit in my example
A school's worth of students. Which you select not at random - but based on which school a student attends.
quote:
Look up the concept of nesting as it relates to randomization and get back to me champ.
Continuing to show off your superficial understanding of statistics isn't going to help you with your fundamental misunderstandings - or, as I feel the case probably is - your inability to admit you were wrong.
quote:
BTW, these are intro level stats questions. The fact you have such difficulty with them is astoundign for a PhD in astrophysics.
You don't even understand what constitutes a random sample.
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