Normal Curve in standard form: Answer each of the following questions

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1 Basic Statistics Normal Curve in standard form: Answer each of the following questions What percent of the normal distribution lies between one and two standard deviations above the mean? What percent of the normal distribution lies above three standard deviations above the mean? If there were 100,000 persons arrayed in a normal distribution of heights, how many would be expected to lie more than three standard deviations above the mean? Note: Z requires population µ and σ to be known!! Which is typically not the case 5/18/2004 EE-547-DOE Terms Steve Brainerd 1

2 EE-547-DOE : Statistical Process Control and Design of Experiments Steve Brainerd Distributions: Frequency Distributions and Histograms Central Limit Theorem: For large sample sizes, the sampling of the mean can be approximated very closely with a normal distribution, no matter what the real distribution of that population. ( standard error of the mean) 5/18/2004 EE-547-DOE Terms Steve Brainerd 2

3 Basic Statistics Z-Score Standard deviation: Z SCORE: Normalize the curve: mean = 0.0. and sigma = 1.0. distribution denoted: N(0,1) Easier to work with. Z is called a transformed statistic. Also called Standardized normal deviate or unit normal deviate and Z-Score: Now can use to compare populations in terms of standard deviation units. These of course have probabilities associated with them! i.e. A is different than B by 5 standard deviations! Is this a large or small difference? KEY IDEA: We will use this in the future to compare populations and to make inferences. 5/18/2004 EE-547-DOE Terms Steve Brainerd 3

4 Basic Statistics Standard deviation: Area under the curve Z=Score: Normalized statistic. Note the use of Z is for comparisons of the sample data mean to the true populations mean and standard deviation. 5/18/2004 EE-547-DOE Terms Steve Brainerd 4

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6 Standard Error: If we took several samples of the same thing, we could compute several means, one for each sample. We can compute the standard deviation of the sample means and use it as an estimate of their variation around the true sample mean. This standard deviation of means is called the Standard Error. We normally only have 1 sample, but we still desire to measure the mean s variability. We compute an estimated standard error as: For the Z statistic SE is calculated as the Population Standard deviation divided by the square root of the sample size. 5/18/2004 EE-547-DOE Terms Steve Brainerd 6

7 We take a sample size of n We calculate a mean value. The error associated with this mean is defined as SE = True population sigma/(square root n). SE is the uncertainty in this mean value. We use this SE for our Statistical tests. Do not confuse with the populations sigma. It is a scaled population sigma based on the sample size taken. Larger sample equals less uncertainty in the sample mean 5/18/2004 EE-547-DOE Terms Steve Brainerd 7

8 Standard normal curve So what does this say? Probability of getting a mean of or greater for a sample size of 10 due to random chance is 11.9% Or about 12% of the time you would normally expect a mean value of or higher based on the sample size of 10! EXCEL: NORMDIST(-1.18,0,1,1) = (or a greater mean value) 5/18/2004 EE-547-DOE Terms Steve Brainerd 8

9 SE: deviation about the sample mean value given the sample size n. It is the uncertainty in this mean value. n = 1 ; Percent = 36% n = 10 ; Percent = 12% n = 20 ; Percent = 5% True mean sample mean As the sample size increases the uncertainty in the sample mean decreases. So when we ask the question: What % of time can we expect a mean value > than the sample? The % decreases as the sample size increases because the mean sigma SE decreases with sample size. 5/18/2004 EE-547-DOE Terms Steve Brainerd 9

10 12% of the time you would normally expect a mean value of or higher for this sample of size 10! Population Mean TRUE MEAN Population Sigma TRUE SIGMA Sample mean Xbar - µ Sample mean Sigma values from Pop mean Sample size SE Z score P value ( NORMDIST ( - Z,0,1,1)) Probability of expecting a mean > SAMPLE MEAN Medium % % % % % 36% It gets harder to have equivalent MEANs as sample size increases Remember we are looking at the sample mean in comparsion to the real population mean! Large % = 1 sigma % % % Small % % % 5/18/2004 EE-547-DOE Terms Steve Brainerd 10

11 REMEMBER THIS IS JUST LOOKING AT MEANS and not individuals. It is a comparison of the sample mean to the population mean. i.e How possible is it that the true mean would be the same value as the mean obtained from the sample? Notice: 1. As the sample size increases, the probability that this specific sample mean is the same as the true mean tested against goes down! 2. The sample mean is a fixed value. 3. The larger that difference between the sample mean and the true mean, the less likely they are the same! 5/18/2004 EE-547-DOE Terms Steve Brainerd 11

12 We are looking at the sample mean and seeing if it is the same as the population mean. For our example we state the Null: Ho : The difference between the two means is 0. We are stating that the difference between the two means = 7.3 is merely due to random chance, since the real difference is 0. The Alternative Hypothesis Hi: Is the difference in the means larger than can be accounted for by chance? 5/18/2004 EE-547-DOE Terms Steve Brainerd 12

13 12% area We calculate a mean based on a sample size of 10. We calculate an standard deviation or error for this mean value. This is called the Standard Error or SE. It is the uncertainty in this mean value since we do not know exactly what it s real value is. SE is calculated as the Population Standard deviation divided by the square root of the sample size. 5/18/2004 EE-547-DOE Terms Steve Brainerd 13

14 Key Ideas: Z score: If n =1 Then Z uses the true populations complete sigma value True mean sample mean SE: Standard error is the expected deviation about the sample mean value given the sample size n. It is the uncertainty in this mean value 5/18/2004 EE-547-DOE Terms Steve Brainerd 14

15 Key Ideas: Examine the Z score: The numerator is the signal. How large is the difference between true mean and new sample mean Notice the denominator term This is called the SE or Standard Error of the sample mean. It is the variation of the test mean for that sample size. So as the Sample size increases the variation in the sample mean decreases. At an infinite sample size the variation about the sample mean is zero. Then it is no longer a sample, but is the real true population mean! 5/18/2004 EE-547-DOE Terms Steve Brainerd 15

16 Basic Statistics Normal Distribution Example confidence intervals 5/18/2004 EE-547-DOE Terms Steve Brainerd 16

17 Basic Statistics Normal Distribution Example confidence intervals Where did this 1.96 come from? It is the +/- Z scores representing the upper and lower limits containing 95% of the sample population data! Can calculate in Excel as: = NORMDIST(Z,1,0,1) = NORMDIST (1.96,1,0,1) = Z NORMSDIST(Z) 1 -NORMSDIST(Z) Z 1-2*(1-((NORMSDIST(Z)))) 2*(1-((NORMSDIST(Z)))) P NORMSINV(P) In the days before computers we had to look up in tables! 5/18/2004 EE-547-DOE Terms Steve Brainerd 17

18 Basic Statistics Normal Distribution Example confidence intervals For our example here: µ = σ = z for 95% confidence interval = 1.96 Value mean std LOWER UPPER z 95% Thus: we are 95% confident that the true mean is contained in the interval = ( *19.64) < µ < ( *19.64) = < µ <190.6 z 99% /18/2004 EE-547-DOE Terms Steve Brainerd 18