1.3 Density Curves and Normal Distributions. Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102

Transcription

1 1.3 Density Curves and Normal Distributions Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102

2 Fitting Density Curves to Histograms Advanced statistical software (NOT Microsoft Excel) can produce smoothed versions of histograms. Example The following are histograms and corresponding density curves for data representing: (a) the acidity or rainwater; (b) the survival time of Guinea pigs.

3 Fitting Density Curves to Histograms When fitting a density curve to a histogram, we want that for any interval on the horizontal axis that spans the width of a collection of rectangles, the following holds: area of rectangles area under density curve. This requirement follows from the more general fact that for both histograms and density curves, area = proportion.

4 Definition of Density Curve A density curve is a curve that is always on or above the horizontal axis and has area exactly 1 underneath it. In addition, we have that for any two values a and b on the horizontal axis, area below the density curve between a and b proportion of observations that fall between a and b.

5 Median of a Density Curve The median of a density curve is the point M on the horizontal axis so that the area below the density curve and to the left of M is 50% (and consequently the area to the right is also 50%) Median

6 Percentiles of a Density Curve The pth percentile of a density curve is the point P on the horizontal axis so that p percent of the area below the density curve lie to the left of P. The inter-quartile range is consequently the extent of the middle 50% of the area. 50 Q 1 Q 3

7 Mean of a Density Curve The mean of a density curve is the balance point of the curve: if the area below the curve were made of a solid material, the mean would correspond to the position of the fulcrum when balancing it:

8 Mean and Median of a Density Curve Unless a density curve is symmetric, the mean is not equal to the median. For right-skewed distributions the mean is larger than the median; For left-skewed distributions the mean is smaller than the median.

9 Normal Distributions Normal curves are the density functions of normal distributions. They have the following general shape. They are symmetric, unimodal (have only one peak), and bell-shaped. The mean is denoted by the symbol µ (small Greek letter mu ), and the standard deviation is denoted by the symbol σ (small Greek letter sigma ). On either side of the mean there are two points, called inflection points where the curve makes the transition from bending upwards to bending downwards, and vice versa. The standard deviation σ is the horizontal distance from the mean µ to these inflection points.

10 Normal Distributions Two normal curves are shown here.

11 The Rule For normally distributed variables, we have the following statements about the relation of the mean/standard deviation and the distribution of the variable: About 68% of all observations lie within one standard deviation of the mean. 68% SD

12 The Rule About 95% of all observations lie within two standards deviation of the mean. 95% SD

13 The Rule About 99.7% of all observations lie within three standards deviation of the mean. 99.7% SD

14 z-scores Recall that a z-score is the number of standard deviations that an actual value lies away from the mean: z = x µ σ. So, if a variable has a normal distribution, we can translate z-scores to positions within the overall distribution, as in the following. About 68% of all observations have a z-score between 1 and 1. About 95% of all observations have a z-score between 2 and 2. About 99.7% of all observations have a z-score between 3 and 3.

15 z-scores Example. According to the collective experience of pediatricians, pregancy durations are normally distributed with a mean of 266 days and a standard deviation of 16 days. Suppose a pregnancy lasts 310 days. The z-score of x = 310 is z = = Thus, a pregnancy of this duration would be very unusual to occur, since a z-score of 2.75 would be outside a normal range of e.g. ±2 standard deviations from the mean.

16 Example: Height of Young Women The height of young women aged 18 to 24 is approximately normally distributed with mean µ = 64.5 inches and standard deviation σ = 2.5 inches. We write X N(µ, σ) if a variable X has a normal distribution with mean µ and standard deviation σ. Consequently, we have that for the height X of young women, X N(64.5, 2.5).

17 Example: Height of Young Women Find the following, using a TI-83/TI-83 Plus/TI-84 Plus calculator: The percentage of women that are between 60 and 65 inches tall. 1. Type [2ND] VARS (DISTR), select 2: normalcdf(.

18 Example: Height of Young Women 2. Type normalcdf(60,65,64.5,2.5) and press ENTER. 3. The proportion is , so the percentage of women who are between 60 and 65 inches tall is about 54.3%. This means, the shaded area is 54.3%. 54.3% Note: The general syntax for finding the proportion between a and b is normalcdf(a,b,µ,σ).

19 Example: Height of Young Women Find the percentage of women who are taller than 62 inches. 1. Type [2ND] VARS (DISTR), select 2: normalcdf(. 2. Type normalcdf(62, ,64.5,2.5). (The number can be replaced by any very large positive number.) 3. Press ENTER. The proportion is , so the percentage of women who are taller than 62 inches is about 84.1%. 84.1%

20 Example: Height of Young Women What is the cutoff score for the top 10% (i.e. the 90th percentile)? 1. Type [2ND] VARS (DISTR), select 3: invnorm(. 2. Type invnorm(0.9,64.5,2.5) and press ENTER.

21 Example: Height of Young Women 3. The percentile is , so 90% of women are shorter than 67.7 inches (and 10% are taller than 67.7 inches). 90% The general syntax for finding the cutoff so that the proportion p of observations fall below this cutoff is invnorm(p,µ,σ).

22 Example: Height of Young Women Find range of the middle 80% of the distribution. This means we need to find the 10th and the 90th percentile. 80% 1. The 90th percentile was computed above, it is Type invnorm(0.1,64.5,2.5) to find the 10th percentile. It is So the middle 80% of the heights ranges from 61.3 inches to 67.7 inches.

23 ios App Use Scroll all the way down to and click on Distribution Calculator. Then, select Normal Distribution.

24 ios App Use The percentage of women who are between 60 and 65 inches tall is about 54.3%.

25 ios App Use The percentage of women who are taller than 62 inches is about 84.1%.

26 ios App Use The 10th percentile is 61.3 and the 90th percentile is 67.7.