The Normal Distribution and Z-Scores

The Normal Distribution The standard deviation
Benefits: Uses measure of central tendency (i.e. mean) Uses all of the data points Has a special relationship with the normal curve Can be used in further calculations 2 Normal Distribution 0.025 0.02 f(X) 0.015 0.01 0.005 0 20 40 60 80 100 120 140 160 180 Example: The Mean = 100 and the Standard Deviation = 20 3 1 Normal Distribution (Characteristics) Horizontal Axis = possible X values
Vertical Axis = density (i.e. f(X) related to probability or proportion)
Defined as
f (X ) = f (Xi ) =
1 σ 2π (e ) − ( X − µ ) 2 2σ 2 2 1 *(2.71828183)− ( X i − X ) ( s ) 2 * (3.14159265) 2 s2 The distribution relies on only the mean and s 4 Normal Distribution (Characteristics)
Bell shaped, symmetrical, unimodal
Mean, median, mode all equal
No real distribution is perfectly normal
But, many distributions are approximately normal, so normal curve statistics apply
Normal curve statistics underlie procedures in most inferential statistics. 5 f(X) Normal Distribution µ + 4sd µ + 3sd µ + 2sd µ + 1sd µ − 1sd µ − 2sd µ − 3sd µ − 4sd µ 6 2 The standard normal distribution What happens if we subtract the mean from all scores?
What happens if we divide all scores by the standard deviation?
What happens when we do both???
7 Normal Distribution 0.025 0.02 f(X) 0.015 0.01 0.005 0 -mean /sd both 20 40 60 80 100 120 140 160 180 -80 1 -4 -60 2 -3 -40 3 -2 -20 4 -1 0 5 0 20 6 1 40 7 2 60 8 3 80 9 4 8 The standard normal distribution
A normal distribution with the added properties that the mean = 0 and the s=1
Converting a distribution into a standard normal means converting raw scores into Z-scores 9 3 Z-Scores
Indicate how many standard deviations a score is away from the mean.
Two components: Sign: positive (above the mean) or negative (below the mean). Magnitude: how far from the mean the score falls 10 Z-Score Formula
Raw score → Z-score X −X score - mean Zi = i = s standard deviation
Z-score → Raw score X i = Zi (s) + X 11 Properties of Z-Scores
Z-score indicates how many SD’s a score falls above or below the mean.
Positive z-scores are above the mean.
Negative z-scores are below the mean.
Area under curve  probability
Z is continuous so can only compute probability for range of values 12 4 Properties of Z-Scores
Most z-scores fall between -3 and +3 because scores beyond 3sd from the mean
Z-scores are standardized scores → allows for easy comparison of distributions 13 The standard normal distribution
Rough estimates of the SND (i.e. Z-scores): 14 The standard normal distribution
Rough estimates of the SND (i.e. Z-scores): 50% above Z = 0, 50% below Z = 0 34% between Z = 0 and Z = 1, or between Z = 0 and Z = -1 68% between Z = -1 and Z = +1 96% between Z = -2 and Z = +2 99% between Z = -3 and Z = +3 15 5 Normal Curve - Area
In any distribution, the percentage of the area in a given portion is equal to the percent of scores in that portion Since 68% of the area falls between ±1 SD of a normal curve 68% of the scores in a normal curve fall between ±1 SD of the mean 16 Rough Estimating Example: Consider a test (X) with a mean of 50 and a S = 10, S2 = 100
At what raw score do 84% of examinees score below?
30 40 50 60 70 17 Rough Estimating Example: Consider a test (X) with a mean of 50 and a S = 10, S2 = 100
What percentage of examinees score greater than 60?
30 40 50 60 70 18 6 Rough Estimating Example: Consider a test (X) with a mean of 50 and a S = 10, S2 = 100
What percentage of examinees score between 40 and 60?
30 40 50 60 70 19 Have→Need Chart When rough estimating isn’t enough Zi = Xi − X s Raw Score Table D.10 Start with Z column Z-score X i = Zi (s) + X Area under Distribution Table D.10 Start with the Mean to Z Column 20 Table D.10 21 7 Smaller vs. Larger Portion Smaller Portion is .1587 Larger Portion is .8413 22 From Mean to Z Area From Mean to Z is .3413 23 Beyond Z Area beyond a Z of 2.16 is .0154 24 8 Below Z Area below a Z of 2.16 is .9846 25 What about negative Z values?
Since the normal curve is symmetric, areas beyond, between, and below positive z scores are identical to areas beyond, between, and below negative z scores.
There is no such thing as negative area! 26 What about negative Z values? Area above a Z of -2.16 is .9846 Area below a Z of -2.16 is .0154 Area From Mean to Z is also .3413 27 9 Keep in mind that…
total area under the curve is 100%.
area above or below the mean is 50%.
your numbers should make sense. your area make sense? Does it seem too big/small?? Does 28 Tips to remember!!! 1. 2. 3. 4. Always draw a picture first Percent of area above a negative or below a positive z score is the “larger portion”. Percent of area below a negative or above a positive z score is the “smaller portion”. Always draw a picture first! 29 Tips to remember!!! 5. 6. 7. Always draw a picture first!! Percent of area between two positive or two negative z-scores is the difference of the two “mean to z” areas. Always draw a picture first!!! 30 10 Converting and finding area
Table D.10 gives areas under a standard normal curve.
If you have normally distributed scores, but not z scores, convert first.
Then draw a picture with z scores and raw scores.
Then find the areas using the z scores. 31 Example #1
In a normal curve with mean = 30, s = 5, what is the proportion of scores below 27? Z 27 = 27 − 30 = −0.6 5 Smaller portion of a Z of .6 is .2743 Mean to Z equals .2257 and -4 -3 -2 -1 0 1 2 3 4 .5 - .2257 = .2743 Portion ≅ 27% 27 32 33 11 Example #2
In a normal curve with mean = 30, s = 5, what is the proportion of scores fall between 26 and 35? Z 26 = .3413 .2881 26 − 30 = −0.8 5 Mean to a Z of .8 is .2881 Z 35 = 35 − 30 =1 5 Mean to a Z of 1 is .3413 .2881 + .3413 = .6294 -4 -3 -2 -1 0 1 2 3 4 Portion = 62.94% or ≅ 63% 26 34 35 Example #3
The Stanford-Binet has a mean of 100 and a SD of 15, how many people (out of 1000 ) have IQs between 120 and 140? Z140 = .4082 140 − 100 = 2.66 15 Mean to a Z of 2.66 is .4961 Z120 = 120 − 100 = 1.33 15 Mean to a Z of 1.33 is .4082 ←.4961→ → .4961 - .4082 = .0879 -4 -3 -2 -1 0 1 120 2 3 140 4 Portion = 8.79% or ≅ 9% .0879 * 1000 = 87.9 or ≅ 88 people 36 12 37 When the numbers are on the same side of the mean: subtract - = 38 Example #4
The Stanford-Binet has a mean of 100 and a SD of 15, what would you need to score to be higher than 90% of scores? In table D.10 the closest area to 90% is .8997 which corresponds to a Z of 1.28 90% IQ = Z(15) + 100 IQ = 1.28(15) + 100 = 119.2 40 55 70 85 100 115 130 145 160 39 13 40 14