Chebyshev's theorem and the empirical rule

Examine the Empirical Rule describing the Normal Curve. Compare how the Empirical Rule and Chebyshev's Theorem describe area under the curve at two and three standard deviations from the mean.

1. What are the differences and why are there differences?

2. In what areas of your daily lives are normal curves used to describe things?

3. What would you say if your local school scored a z = 0 on the fourth grade reading achievement test?

4. Let's look at sampling distributions. Why will a sampling distribution yield a better view of the population than a single sample? How might a single sample differ from a sampling distribution?

5. Sampling size is something else that we need to consider. At what point in sampling size can we assume that a normal distribution will be seen? Why is this number considered to be one of statistic's "magic" numbers?

Respond:

Examine the Empirical Rule describing the Normal Curve. Compare how the Empirical Rule and Chebyshev's Theorem describe area under the curve at two and three standard deviations from the mean. What are the differences and why are there differences?

Empirical rule states that “for data sets having a distribution that is approximately bell-shaped” : 95% of all values are within 2 std deviations and 99.7% within 3.

Chebyshev’s theorem can apply to bell shaped curves but unlike the Empirical rule can be used additionally for any data set. The penalty here is a reduction in certainty as the approximations for a Chebyshev graph at 2 std deviations is only 75% and just 89% at 3.

In what areas of your daily lives are normal curves used to describe things?

I suppose you can graph out your time awake (or time asleep for that matter). I’m not very concerned with the amount of chocolate chips or volume of Coke so while there are things that can be measured, what do I actually use a normal curve for??? How about energy usage in the home? I can’t say I look at any graphs of my usage data, but I suppose it’s there broken by month week or probably day. I guess if I had a high electrical bill compared to the “mean” of previous months I could look up the curves to figure out where I was using more power.

What would you say if your local school scored a z = 0 on the fourth grade reading achievement test?

I would ask a 5th grader and see if I were smarter….or more likely just say “Yay for mediocrity!” You’re only as good as the next 4th grader around. Way to set the bar high kid…”

Let's look at sampling distributions. Why will a sampling distribution yield a better view of the population than a single sample? How might a single sample differ from a sampling distribution?

We know that we have to have certain minimum sample sizes to obtain a population accurate statistic. An individual sample could be an outlier or far off the norm, but by using a sampling distribution on even skewed population data tends to have a normal distribution of sample means.

Sampling size is something else that we need to consider. At what point in sampling size can we assume that a normal distribution will be seen? Why is this number considered to be one of statistic's "magic" numbers?

30 seems to be the magic number for such a distribution, as seen in the Central Limit Theorem but in truth if your sample already shows a tend to normal distribution smaller samples can work. Conversely, the further skewed the data is the larger the ‘n’ sample size needs to be to normalize it.