1. The Normal Curve and Standard Scores
Chapter 5
The Normal Curve and Standard Scores

2. Quiz Which set of data would have the highest standard deviation?b) c)

3. Chapter 5: The Normal Curve
Many variables are distributed normally in nature and in business.
This means there is a “normal” value.
Example: The height of women.
Example: Hours worked per year by full time employees.
Many dependent variables (DVs) are distributed normally in experiments.
The more data you have, the smoother the curve gets.

4. The Normal Curve

5. Areas under the normal curve
Example: IQs: mean = 100, SD = 16

6. Standard Scores (z-scores)
Suppose a child’s IQ is What percentile of intelligence is she in?

7. Standard Scores (z-scores)
If a child’s IQ is 132, we could also say she was two standard deviations above the mean. This would mean she has a z-score of 2
General formula:
z-scores allow us to compare the magnitude of scores that are not normally comparable (shoe size to height, or stats grade to English grade).
They give us a measure of how extreme a score is (e.g., commute time).

8. Calculating z-Scores 1. Calculate mean and standard
deviation (data analysis)
2. Calculate a z-score for each
raw score: z = (X – μ)/σ
In our example: z =(a2-$e$5)/$e$9
Use $ for “absolute cell reference”

9. Characteristics of z-scores
1. z-scores can be calculated for distributions of any shape, not just normal distributions.
2. The mean of the z-scores for a population is always 0.
3. The standard deviation of the z-scores for a population is always 1.

10. Standard Scores IQ scores and the normal curve
What percentage of people have an IQ below 116?
50% % =84.13%

11. What to do for z scores other than +/-1,2,3
What percentage of children have an IQ under 142?
Solution: Convert data into z-scores and use the normsdist formula in Excel to get the area under the curve to the left of z.

12. NORMSDIST function Normal Standardized Distribution
=normsdist(z) gives us the area to the left of z
= fraction of scores below z
= probability of getting a score lower than the one associated with z

13. What to do for z scores other than +/-1,2,3
What percentage of children have an IQ under 142?
Solution: Convert data into z-scores and use the normsdist formula in Excel to get the area under the curve to the left of z.
z = (X– μ)/σ = (142 – 100)/16 = 2.63

14. Chips at Taco King Brittany: 0 Loren 3 Rochelle 8 Bree 11 Paris 22
According to the lady who works there, the average customer eats 8 chips, and most eat between 3 and 13 (standard deviation = 5)
Calculate z for each person.

15. Chips at Taco King Chapter 5 Chips Data.xslx
Use z = (X – Mean)/SD to calculate z-score.
Excel formula top cell: =(E5-$E$11)/$E$12
Change number format to “Number” so only 2 decimals show.

16. NORMSDIST function Normal Standardized Distribution
=normsdist(z) gives us the area to the left of z
= fraction of scores below z
= probability of getting a score lower than the one associated with z

17. Chips at Taco King Chapter 5 Chips Data.xslx
Fraction to left of a z score = NORMSDIST(z)
Excel formula top cell: =normsdist(f5)

18. NORMSDIST function Normal Standardized Distribution
Percentile associated with a z-score
=100*normsdist(z)

19. Chips at Taco King Chapter 5 Chips Data.xslx
Percentile of a z score = NORMSDIST(z)*100
Excel formula top cell: =normsdist(f5)*100

20. NORMSDIST function Normal Standardized Distribution
Area to the right of a z score
= fraction of population with a higher score
=1 - normsdist(z)

21. Chips at Taco King Chapter 5 Chips Data.xslx
Area to the right of a z score = 1 – Area to the left
Excel formula top cell: = 1 - normsdist(f5) OR
= 1 – g5

22. Chips at Taco King
1. What percentage of people eat as many or more chips than Bree?
2. What percentage of people eat as many or more chips than Paris?
3. What percentage of people eat as few or fewer chips than Brittany?
4. Rochelle is in what percentile of chip eaters?
5. Bree is in what percentile of chip eaters?

23. How to Calculate the Area Under the Normal Curve Between 2 Scores
Convert the scores to z-scores
The area between two z-scores, a (smaller) and b (larger)
=normsdist(b) – normsdist(a)

24. Chips at Taco King Chapter 5 Chips Data.xslx
Area between to z-scores
=Fraction to Left(high score) – Fraction to the Left(low score)
= normsdist(high score) – normsdist(low score)
What fraction of
People eat more than
Loren but less than
Rochelle?
= Fraction to the left (Rochelle) –
Fraction to the left (Loren) =
= 0.34 = 34%

25. Calculating a Raw Score that Meets a Criteria.
How many chips would someone need to eat to be in the in the 75th percentile?
Area to left of z = .75
Use NORMSINV(area) to find z.
NORMSINV(probability) is conceptually the same.
Excel command to find z: = normsinv(.75)
If z = (X – Mean)/SD, then X = Mean +z*SD

26. T-Scores
Since z-scores are always small numbers, sometimes they are transformed into T-Scores (μ = 50, σ = 10).
Especially used in evaluations, education, and lab results
T-score = 50 + z*10
Examples:
z = 1 T = 60
Z = 0 T = 50
z = -.5 T = 45
Other transformed score types.
IQ score = z*16 or z*15
SAT type score = z*100

27. Normalized Weighted Averages Chapter 5 Candidate Data.xlsx
If you’re creating a weighted average, and if the variances (SDs) are different for the criteria, you need to use normalized scores (z-Scores, T-Scores, etc.).

28. Normalized Weighted Averages Chapter 5 Candidate Data.xlsx
Standardized scores give a more accurate picture since they all have the same mean and variance (SD).

29. Beanie Babies
μ =125 g
σ = 15 g

30. Beanie Baby Exercise
Calculate the z-score for the weight of each Beanie Baby.
1. What percentage of all Beanie Babies weigh more than your heaviest Beanie Baby?
2. What percentage of all Beanie Babies weigh less than your lightest Beanie Baby?
3. What percentage of all Beanie Babies are below the median weight of your Beanie Babies?
4. What percentage of Beanie Babies have weights more extreme than your Beanie Babies?
5. What percentage of all Beanie Babies have weights between your two heaviest Beanie Babies?