University:
California State University, Northridge
Course:
MATH 140 | Introductory Statistics
Academic year:

2010
Views:

495

Pages:

23
Author:

Kyleigh Robbins

Math 140 Introductory Statistics Chapter 2 Visualizing Distributions Recall the definition: The values of a summary statistic (e.g. the average age of the laid-off workers) and how often they occur. Four of the most common basic shapes: Uniform or Rectangular Normal Skewed Bimodal (Multimodal) Uniform (or Rectangular) Distribution Each outcome occurs roughly the same number of times. Examples. Number of U.S. births per month in a particular year (see Page 25) Computer generated random numbers on a particular interval. Number of times a fair die is rolled on a particular number. Births Deaths (in thousands) (in thousands) 1 305 218 2 289 191 3 313 198 4 342 189 5 311 195 6 324 182 7 345 192 8 341 178 9 353 176 10 329 193 11 304 189 12 324 192 Month Uniform (or Rectangular) Distribution Month Births Deaths (in thousands) (in thousands) 1 305 218 2 289 191 3 313 198 4 342 189 300 5 311 195 200 6 324 182 7 345 192 8 341 178 9 353 176 10 329 193 11 304 189 12 324 192 Births in US (1997) 100 11 8 5 0 1 Number in Thousands 400 Month Births Normal Distributions These distributions arise from Variations in measurements. (e.g. pennies example, see 2.3 page 31) Natural variations in population sizes (e.g. weight of a set of people) Variations in averages of random samples. (e.g. Average age of 3 workers out of 10, see 1.10 in page 14) Pennies example Average age of 3 workers out of 10 Normal Distributions Idealized shape shown below (see 2.4 page 32) Properties: Single peak: The x-value of it is called the mean. The mean tells us where is the center of the distribution. The distribution is symmetric with respect to the mean. Mean Normal Distributions Idealized shape shown below (see 2.4 page 32) Properties: Inflection points: Where concavity changes. Roughly 2/3 of the area below the curve is between the inflection points. Inflection Points Mean Normal Distributions Idealized shape shown below (see 2.4 page 32) Properties: The distance between the mean and either of the inflection points is called the standard deviation (SD) The standard deviation measures how spread is the distribution. SD SD Mean Skewed Distributions These are similar to the normal distributions but they are not symmetric. They have values bunching on one end and a long tail stretching in the other direction The tail tells you whether the distribution is skewed left or skewed right. Skewed Left Skewed Right Skewed Distributions Skewed distributions often occur because of a “wall”, that is, values that you cannot go below or above. Like zero for positive measurements, or 100 for percentages. To find out about center and spread it is useful to look at quartiles. Skewed Left Skewed Right Example of a skewed right distribution Median and Quartiles Median: the value of the line dividing the number of values in equal halves. (Or the area under the curve in equal halves.) Repeat this process in each of the two halves to find the lower quartile (Q1) and the upper quartile (Q3). Q1, the median, and Q3 divide the number of values in quarters. The quartiles Q1 and Q3 enclose 50% of the values. Visualizing Median and Quartiles Bimodal Distributions. Previous distributions have had only one peak (unimodal) but some have two (bimodal) or even more (multimodal). Bimodal Distribution Example of a bimodal distribution Using the calculator (TI-83) How to generate a list of n random numbers between 0 and 1 (exclusive). Example: Generate 5 random numbers between 0 and 1. MATH PRB 1.randInt( Enter 5) Enter How to store a list of numbers. Example: Store the previous list of 5 random numbers between 0 and 1 on L1. 2nd ANS → 2nd L1 Using the calculator (TI-83) Example: Store the list 1,2,3,4,5 to L1. STAT 1.Edit Enter Move to the first row of column L1 using the arrows. Type each of the numbers on the list followed by ENTER. Compute binomial coefficients. Example: Compute 10 choose 3. 10 MATH PRB nCr Enter 3 Practice P3. For each of the normal distributions in below, estimate the mean and standard deviation visually, and use your estimates to write a verbal summary of the form “A typical SAT score is roughly (mean), give or take (SD) or so.” Mean ~ 500 SD ~ 100 A typical SAT score is roughly 500, give or take 100 or so. Mean ~ 20 SD ~ 5 A typical ACT score is roughly 20, give or take 5 or so. Practice P4. Estimate the median and quartiles for the distribution of GPAs in Display 2.7 on page 34. Then write a verbal summary of the same form as in the example. 2.9 Lower quartile ~ 2.9 Median ~ 3.3 Upper quartile ~ 3.7 3.3 3.7 The middle 50% of the GPAs of statistic students were between 2.9 and 3.7, with half above 3.3 and half below. Practice P5. Match each plot in Display 2.14 with its median and quartiles (the set of values that divide the area under the curve into fourths). a. 15, 50, 85 b. 50, 71, 87 c. 63, 79, 91 d. 35, 50, 65 e. 25, 50, 75 IV II V III I