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Introduction

Introduction. Population – the entire group of concern Sample – only a part of the whole Based on sample, we’ll make a prediction about the population. Bad sampling: convenience, bias, voluntary Good sampling: simple random sample(SRS). Inferential Stats: making predictions or

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Introduction

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  1. Introduction Population – the entire group of concern Sample – only a part of the whole Based on sample, we’ll make a prediction about the population. Bad sampling: convenience, bias, voluntary Good sampling: simple random sample(SRS). • Inferential Stats: making predictions or • inferences about a population based on a sample

  2. Experiments Observation – no attempt to influence Experiment– deliberately imposes some treatment Basic design principles: Control the effects of lurking variables Randomize which subject gets which treatment Use large sample size to reduce chance variation • Statistical Significance: • An observed effect so big that it would rarely • occur just by chance.

  3. Picturing Distributions with Graphs What makes up any set of data? • Individuals • objects described by data • can be • Variables • characteristic of individuals of particular interest • different values possible for different people

  4. Two kinds of variables • Categorical (Qualitative) • describes an individual by category or quality. • examples like • Numerical (Quantitative) • describes an individual by number or quantity. • discrete for variables that are • continuous for variables that are • examples like

  5. Describing Categorical Variables Tables summarize the data set by • listing possible categories. • giving the number of objects in each category. • or show the count as a percentage. Picture the distribution of a cat. var. with • Pie charts • Bar graphs

  6. Pie Chartswhole is split into appropriate pieces.

  7. Bar Graph Horizontal line keeps track of categorical values. Vertical bars at each value keeps track of # or %. % # 25 20 15 12 5 4 A B C D E F

  8. Example 1 80 AASU students in an Elem. Stats class come from one of four colleges (S & T, Edu, Health, Lib. Arts). The breakdown of these 80 students is given below.

  9. Ex1 - Pie Chart

  10. Ex1 – Bar Graph % 30 20 10 U LA E H ST

  11. Describing Quantitative Variables Tables summarize the data set by • listing possible intervals (ranges, classes). • giving the number of individuals in each class • or showing the number as a percentage. Picture the distribution of a quant. var. with • Histogram (similar to bar graph but now vertical • bars of neighboring classes touch) • Where one class ends, the next begins.

  12. Example 2 Consider the ages of the full-time faculty in the math dept. The breakdown of these 19 individuals is given in the table. % 30 20 10 10 30 50 70

  13. Info from histograms Helps to describe a distribution with • pattern (shape, center, spread) • deviations (outliers) from the rest of the data • Could result from unusual observation or typo • For shape, look at symmetric vs. skewed

  14. Examples 3 and 4 % 2 4 6 8 10 12 % v 40 60 20 80 100

  15. Example 4 without outliers % 30 10 5 v 20 40 60 80 100 % 20 10 5 v v v 20 40 60 80 100

  16. Describing Distributions with Numbers There are better ways to describe a quantitative data set than by an estimation from a graph. Center: mean, median, mode Spread: quartiles, standard deviation

  17. Center: Mean • The mean of a data set is the arithmetic average of • all the observations. • Given a data set:

  18. Mean – Example 1 • Your test scores in a Stats Class are: 60, 75, 92, 80 • Your mean score is:

  19. Mean – Example 2 • Compare high temperatures in Savannah for July 2010 and July 2011. • July 2010 high temps: 83, 87, 84, …, 97, 100, 92 • July 2011 high temps: 94, 91, 93, …, 97, 99, 99

  20. Center: Median • The median of a data set is the middle value of • all the (ordered) observations. • Given a data set:

  21. Median – Examples 3/4 • 11 tests: 60, 77, 92, 80, 84, 93, 80, 95, 65, 66, 75 • Ordered data set: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95 • 10 dice rolls: 2, 4, 5, 5, 6, 7, 7, 8, 9, 10

  22. Center: Mode • The mode of a data set is the value that appears the most. • Tests data set: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95 • Dice rolls: 2, 4, 5, 5, 6, 7, 7, 8, 9, 10 • 2010 July High Temps mode: • 2011 July High Temps mode:

  23. Spread: Quartiles A measure of center is not useful by itself • Are other observations close or far from center? Take an ordered data set and find: • M, • Q1, • Q3, • IQR = Summary of data in the “Five-Number Summary”:

  24. Quartiles – Example 5 • 11 tests: 60, 65, 66, 75, 77, 80, 80, 84, 92, 93, 95 • 5-num-sum: • Visualize 5-num-sum with a boxplot. • Draw rectangle with ends at Q1 and Q3. • Draw line in the box for the median. • Draw lines to the last observations within 1.5IQR of the quartiles. • Observations outside 1.5IQR of the quartiles are suspected outliers.

  25. Boxplot – Example 6 • 5-Num-Sum: 60, ____, 80, ____, 95 50 60 70 80 90 100 • Draw rectangle with ends at Q1 and Q3 • Draw line in the box for the median • Draw lines to last observations within 1.5IQR of the quartiles • Observations outside 1.5IQR of the quartiles are suspected outliers

  26. Boxplot – Example 7 • July 2010 5-Num-Sum: 83, 92, 94, 97, 102 • July 2011 5-Num-Sum: 84, 91, 95, 98, 99 • 2010 • IQR = 97-92=5 2010 2011 • 2011 • IQR = 98-91=7 80 85 90 95 100 105

  27. Spread: Standard Deviation More common measure of spread (in conjunction with the mean) is the standard deviation. A single deviation from the mean looks like For every value in a data set, deviations are either positive, negative or zero. Finding an average of those will be trouble, since when you add the deviations together, you’ll get 0. • Example 1 data: 60, 75, 92, 80

  28. To deal with this “adding to zero”, we get rid of any negative terms by squaring each deviation. A single squared deviation from the mean looks like: The average of the squared deviations is called the variance: • n-1 is called the degrees of freedom, since knowledge of the first (n-1) deviations will automatically set the last one.

  29. The standard deviationis the square root of the variance.

  30. When to use what? For skewed data: For (nearly) symmetric data: Outliers have a big impact on mean and std. dev. Consider two data sets: • Set 1: 1, 1, 3, 5, 10 • Set 2: 1, 1, 3, 5, 70

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