Chebyshev's rule / theorm --> help with statistics problem?

Question

Okie, the problem is this:

I have a histogram, positively skewed.

The book says the mean is 27 and the standard deviation is 24.

The book asks to find the number of observation within 0 to 75 and 0 to 47.

The whole range of the histogram is only 0 -200, and cannot go below zero since that would be negative time.


Can anyone help out?

we're supposed to use chebyshev's rule Thanks.

Thanks Mr.DJ, but we are to use chebyshev's rules to solve this.

Yes, I understand that the equation is:

100*(1-(1/k^2))

BUT the thing is.

I understand that SD of 2 = 75%, SD 3 =89%

BUT

as you've stated, the lowest the graph goes is to 0. two S.D would equal 24*2 = 48 then subtracted from mean of 27 = -21, which is clearly impossible time wise (min)

but you're saying that it is okay?

To: insolver

You forgot though, because chebyshev's rule is for non-normal curves, that you cannot simply divide 25% by two to get each side like a normal curve, since both sides might not be equal.

To Aldo : Thank you.

Your answer was what I was looking for to confirm my own answer.

I am aware that k does not have to be an integer, but that doesn't really apply here...

Thank you I will be giving you stars... if I know how.. I'm new at this

Answer 1

I am sure you have the equation of the Chebyshev's but it is worth mentioning it here as 1-(1/k^2) where k is a number greater than 1 but not necessarily an integer. Note at k=1 the value reduces to zero.

Also, it is important to remember that this rule applies to ANY sample of measurements regardless of the shape of the frequency distribution. So this applies to your data even if it is positively skewed.

Also, the rule per above expression states that within 2 SD of the mean 27, the distribution will contain 75% of the data. And within 1 SD, few or virtually zero. For 3 SD 8/9 or 88.89 % of data. Compare this with empirical rule of about 68%, 95%, and 99% for 1SD, 2SD, and 3SD respectively.

Now, the observation between 0-75 translates to the mean + 2SD or 27 + 48 = 75. Thus you expect that it will contain 75% of the data in the range 0-75. Remember within 1 SD, it is essentially zero; and since 47 is within or less than 27 + 24 = 51, that is within one SD, then it is 0%.

The response of Statistic above is incorrect of using 1.4SD because this corresponds to the range of values 33.6 + 27 or the range from 0-60 but you are asking for the range 0-47.

Answer 2

When a distribution is not normal, we use Chebyshave's rule. This states that atleast 1-1/k^2 proportion of data falls within +/-k standard deviations. 75 is (75-27)/24 =2 standard deviations away from the mean. So K can be taken as 2. Then 1-1/k^2 will be 75%.
You suggest there is no negative value. Hence I would think half of the total tail area of 25% ie 12.5% will be outside the zone of interest. So 87.5% of the observations will fall within 0 to 75%. I am not sure about the second range. The k value here is less than 1. I do not think this range would be applicable for this rule.
I should confess, I am not speaking from competence. I am just thinking aloud and keen to know the right answer myself.

Answer 3

Chebyshev's rule says that for ANY distribution,
1. At least 3/4 of the observations are within 2 SDs from mean
2. At least 1/2 of the observations are within 1.4 SDs from mean

So, at least 3/4 obs are within (27- 2*24) to (27+2*24)
The distribution has a lower bound of zero,
at least 3/4 obs are within 0 to 75

Similarly, at least 1/2 obs are within (27 -1.4*24) to (27 +1.4*24)
ie 0 to 47

Answer 4

NIce work statistic. I was reviewing from my textbook. I've never been very good at Chebyshev.

So, what you've said is that in the equation:

P(|W-mu|>e) = sigma^2/e^2 ,
for the 75% inclusion, you take e = 2*sigma so that you're finding the prob. that an observation, W, is more than 2 sigma from the mean is 0.25

For the 50% guy,
sigma^2/e^2 = 1/2 -> e = sqrt(2)*sigma = 1.414*sigma

Thanks from me!

Answer 5

if that is the case, you may use the Central limit theorem, that is, make it standardized so that the range will be small