1. Density Curves and Normal Distributions
Section 2.1

2. Sometimes the overall pattern of a distribution can be described by a smooth curve. This histogram shows the distribution of vocab scores. We could use it to see how many students scored at each value, or what percent of students got 4’s, above 10, etc.

3. Density Curves
A density curve is an idealized mathematical model for a set of data.
It ignores minor irregularities and outliers

4. Density Curves
Eyes respond to the areas of the bars in a histogram. Bar areas represent proportions of the observations.
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5. Density Curves
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Now look at the curve drawn through the bars. This line is called a Density curve. The area under the curve to the left of 6.0 is shaded. The area is Only 0.01 away from the histogram result. Good approximation.
0.303
0.293

6. Always on or above the horizontal axis
Density Curve
Always on or above the horizontal axis
Has an area of exactly 1 underneath it

7. Types of Density Curves
Normal curves
Uniform density curves
Later we’ll see important density curves that are skewed left/right and other curves related to the normal curve

8. Density Curve
 Area = 1, corresponds to 100% of the data

9. What would the results look like if we rolled a fair die 100 times?
Press STAT ENTER
Choose a list: highlight the name and press ENTER.
Type: MATH  PRB 5:randInt(1,6,100) ENTER
Look at a histogram of the results: ND Y= ENTER
Press WINDOW and change your settings
Press GRAPH. Use TRACE button to see heights.
Let them work this out. Then they sketch. Theirs will not necessarily look the same!

10. What would the results look like if we rolled a fair die 100 times?
Outcomes
30% or 0.3
20% or 0.2
10% or 0.1
Relative Frequency
Let them work this out. Then they sketch. Theirs will not necessarily look the same!

11. In a perfect world…
The different outcomes when you roll a die are equally likely, so the ideal distribution would look something like this:
An example of a uniform density curve.
Example of Uniform density curve.

12. Other Density Curves
What percent of observations are between 0 and 2? (area between 0 and 2)
Area of rectangle: 2(.2) = .4
Area of triangle: ½ (2)(.2) = .2
Total Area = = .6 = 60%

13. Other Density Curves What percent of observations are between 3 and 4?
Area: (1)(.2) = .2 = 20%

14. Normal curve

15. Density Curves: SkewedM
Median: the equal-areas point of the curve
Half of the area on each side
Mark where you think the Median would be.

16. Density Curves: Skewed
Mean: the balance point of the curve (if it was made of solid material)
Mark where you think the mean would be.

17. Mean and Median Of Density Curves
Just remember:
Symmetrical distribution
Mean and median are in the center
Skewed distribution
Mean gets pulled towards the skew and away from the median.

18. Notation
Since density curves are idealized, the mean and standard deviation of a density curve will be slightly different from the actual mean and standard deviation of the distribution (histogram) that we’re approximating, and we want a way to distinguish them

19. Notation For actual observations (our sample): use and s.
For idealized (theoretical): use μ (mu) for mean and σ (sigma) for the standard deviation.

20. Normal Curves are always:
Described in terms of their mean (µ) and standard deviation (σ)
Symmetric
One peak and two tails

21. Normal Curves
Concave down
Inflection point σ
Concave up
Inflection points – points at which this change of curvature takes place.

22. Normal Curves
Larger std. dev. = more spread out, Smaller std. dev. = more pulled in

23. The Empirical Rule
The Rule

24. The Empirical Rule
68% of the observations fall within σ of the mean µ.
68 % of data

25. The Empirical Rule 95% of the observations fall with 2σ of µ.
95% of data

26. The Empirical Rule 99.7% of the observations fall within 3σ of µ.
99.7% of data

27. Heights of Young Women
The distribution of heights of young women aged 18 to 24 is approximately normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
64.5 – 2.5 = 62
= 67
Height (in inches)

28. Heights of Young Women
The distribution of heights of young women aged 18 to 24 is approximately normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches. 5
Height (in inches)
Height (in inches)

29. Heights of Young Women
The distribution of heights of young women aged 18 to 24 is approximately normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
99.7% of data
Height (in inches)
Height (in inches)

30. Shorthand with Normal Dist.
Ex: The distribution of young women’s heights is N(64.5, 2.5).
What this means:
Normal Distribution centered at µ = 64.5 with a standard deviation σ = 2.5.

31. Heights of Young Women 50%
What percentile of young women are 64.5 inches or shorter?
50%
99.7% of data
Height (in inches)

32. Heights of Young Women 2.5%
What percentile of young women are 59.5 inches or shorter?
2.5%
99.7% of data
Height (in inches)

33. Heights of Young Women 64.5 or less = 50% 50% – 2.5% = 47.5%
What percentile of young women are between 59.5 inches and 64.5 inches?
64.5 or less = 50%
59.5 or less = 2.5%
50% – 2.5% = 47.5%
99.7% of data
Height (in inches)

34. Practice
For homework:
2.1, 2.3, p. 83
2.6, 2.7, p. 89