This document provides an overview of z-scores and the normal distribution. It defines key terms like mean, median, mode and standard deviation. It presents the formulas for calculating z-scores and discusses how to find the percentage of data that falls below or above a given z-score using normal distribution tables. Examples are provided to demonstrate how to calculate z-scores and interpret them on the normal distribution curve.

1. Z scores And whatnot

2. This is a NORMAL DISTRIBUTION Most people are near the hump So the Mean, Median, and Mode fall in the middle too Note the symmetry off the curve

3. 0 1 2 3 -1 -2 -3 Normal Distributions standardize information from different data sets so the Mid-point is 0 Each point is just like standard counting numbers, so to speak The segments on the left are negative

4. 0 1 2 3 -1 -2 -3 Its essentially a number line You typically won’t see higher than 4

5. Next we’ll start getting into numbers

6. Here are the formulas… Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) Learn them, Love them

7. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) A problem may be given and from it you can pull out X (some score), Mu (the mean), and Sigma (standard deviation) Here are some numbers for example X = 300 μ = 200 σ = 50 You pretty much just plug in the numbers from there Remember these numbers, I’ll refer to them a lot

8. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) X = 200 μ = 300 σ = 50 Chose a formula that has the matching components Z = X - μ σ

9. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) X = 300 μ = 200 σ = 50 Z = X - μ σ

10. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) X = 300 μ = 200 σ = 50 Z = 300 - 200 50 Solve

11. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) X = 300 μ = 200 σ = 50 Z = 300 - 200 50 100 2

12. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) X = 300 μ = 200 σ = 50 Z = 2 Lets bring back that normal distribution now 0 1 2 3 -1 -2 -3 You put the Z score right on the graph

13. And that’s it

14. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) You can practice with these numbers X= 50 μ =30 σ =5 X= 8 μ =6 σ =2 X =80 μ =100 σ =5 X =18 μ =36 σ =6 4 1 -4 -3 (click for answers)

15. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) You might be wondering about this formula In that case you’d be given Z and solve for X

16. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) Z = 1 μ = 10 σ = 2 You would be given this X = μ + (Z)( σ ) X = 10 + (1)(2) X = 10 + 2

17. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) X = 1 2

18. Easy enough

19. A few more things about the normal distribution 0 1 2 3 -1 -2 -3 There can be more than one Z score on the graph It can be negative And it can have decimals, in that case you just estimate where it sits

20. Z = Z = X - μ σ X - μ S X = μ + (Z)( σ ) X= 50 μ =30 σ =5 X= 8 μ =6 σ =2 X =80 μ =100 σ =5 X =18 μ =36 σ =6 4 1 -4 -3 Remember these answers from before?

21. 4 1 -4 -3 0 1 2 3 -1 -2 -3 Its uncommon to go beyond 3, but it is possible as is the case here -4 4 These numbers were from different data sets but can share the same graph

22. Ok, so what if you’re asked to find a percent of data under or above a certain score?

23. Lets bring back the data we worked with earlier to save some time What if you need to know what percent is below the Z-score? ? X = 300 μ = 200 σ = 50 Z = 2 0 1 2 3 -1 -2 -3

24. Well, at this point you go into that nifty page at the back of your Stat-psych book and look up under column (A) the Z score “2”

25. You can look it up yourself if you want, its on page 529

26. You go to your z score, in our case its 2, then you take the number from the column of the area you want to know. (B) Is body (c) Is tail (D) Is proportion between the Mean and Z

27. This is the body 0 1 2 3 -1 -2 -3

28. This is a tail 0 1 2 3 -1 -2 -3

29. We want the body 0 1 2 3 -1 -2 -3

30. That gives us the number .9772

31. What does that mean!?!

32. Nothing when you want the percent

33. So you just multiply that number (.9772) by 100 and you have your percent of area.

34. 98% .9772 x 100 = 97.72 So about 98% 0 1 2 3 -1 -2 -3

35. Ok, that’s awesome when you want all the data under or above a Z-score, but what if you want to see… lets say, 60% of the data

36. Hmmmm… No picture for this yet Whatever, I’ll skip it 0 1 2 3 -1 -2 -3

37. So you bust out that stylish book of yours and go to page 528 We want 60% of the data so we’re going to look at column D, the proportion between the mean and Z-score and you’re going to want the number closest to .3000

38. Ok, now we can use the picture So, You want to look up the closest number to 30% or .3000 If you look up the closest to .6000 (60%) in column D it will be looking for 60% on both sides And that just doesn’t work That will put 30% on each side adding up to a total 60% 0 1 2 3 -1 -2 -3

39. The number happens to be .2995 Z-score of 0.84 So, lets draw that on 0 1 2 3 -1 -2 -3

40. That’s what it looks like, but now what if you need the score to be within the 60% of the data?

41. X = μ + (Z)( σ ) Remember this formula?

42. Just plug in your numbers and you’ll have your answer

43. X = μ + (Z)( σ ) μ = 200 σ = 50 Z = + .84 Data from before There's the low and the high scores to be within 60% of the data X = 200 + (.84)(50) X = 200 + 42 X = 242 X = 200 + (-.84)(50) X = 200 - 42 X = 158

44. I’ll wrap this up with a quick review of the Central limit theorem as It was taught on Tuesday and I’m not really sure whether I’m suppose to include it or not in my lesson

45. Its for small samples, but you generally want a sample in the 20s or so, a “n” of 3 doesn’t really work so well And you know to use this when you see “n” which we haven’t seen at all in Z-scores

46. So here is the formula Z = M - μ σ m

47. You’ll be given μ, M , σ , and n so it really makes things simple. It only looks complicated but its really quite easy Just take it one step at a time.

48. μ = 95 M = 100 σ = 20 N = 25 Take your numbers and plug them in M - μ 100 – 95 = 5 σ m = σ N √ = 20 √25 20 5 = = 4 4 Z = 1.25 = Z M - μ σ m 1. (Here is the formula) 2. (start with the top) 3. (next do the bottom) 4. (Put it together) = 5 5. (that’s all) Z = M - μ σ m

49. And there you have it

50. Z scores And whatnot