Types of Distributions

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Chapter 1 3
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Statistics

• the science of collecting, analyzing, and drawing
  conclusions from data

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Descriptive statistics

• the methods of organizing summarizing data

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Inferential statistics

• involves making generalizations from a sample to
  a population

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Population

• The entire collection of individuals or objects
  about which information is desired

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Sample

• A subset of the population, selected for study in
  some prescribed manner

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Variable

• any characteristic whose value may change from
  one individual to another

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Data

• observations on single variable or simultaneously
  on two or more variables

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Types of variables
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Categorical variables

• or qualitative
• identifies basic differentiating characteristics
  of the population

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Numerical variables

• or quantitative
• observations or measurements take on numerical
  values
• makes sense to average these values
• two types - discrete continuous

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Discrete (numerical)

• listable set of values
• usually counts of items

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Continuous (numerical)

• data can take on any values in the domain of the
  variable
• usually measurements of something

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Classification by the number of variables

• Univariate - data that describes a single
  characteristic of the population
• Bivariate - data that describes two
  characteristics of the population
• Multivariate - data that describes more than two
  characteristics (beyond the scope of this course

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Identify the following variables

1. the income of adults in your city
2. the color of MM candies selected at random from
   a bag
3. the number of speeding tickets each student in AP
   Statistics has received
4. the area code of an individual
5. the birth weights of female babies born at a
   large hospital over the course of a year

Numerical
Categorical
Numerical
Categorical
Numerical
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Graphs for categorical data
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Bar Graph

• Used for categorical data
• Bars do not touch
• Categorical variable is typically on the
  horizontal axis
• To describe comment on which occurred the most
  often or least often
• May make a double bar graph or segmented bar
  graph for bivariate categorical data sets

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Using class survey datagraph birth month
graph gender handedness
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Pie (Circle) graph

• Used for categorical data
• To make
• Proportion 360
• Using a protractor, mark off each part
• To describe comment on which occurred the most
  often or least often

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Graphs for numerical data
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Dotplot

• Used with numerical data (either discrete or
  continuous)
• Made by putting dots (or Xs) on a number line
• Can make comparative dotplots by using the same
  axis for multiple groups

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Distribution Activity . . .
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Types (shapes)of Distributions
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Symmetrical

• refers to data in which both sides are (more or
  less) the same when the graph is folded
  vertically down the middle
• bell-shaped is a special type
• has a center mound with two sloping tails

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Uniform

• refers to data in which every class has equal or
  approximately equal frequency

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Skewed (left or right)

• refers to data in which one side (tail) is longer
  than the other side
• the direction of skewness is on the side of the
  longer tail

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Bimodal (multi-modal)

• refers to data in which two (or more) classes
  have the largest frequency are separated by at
  least one other class

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How to describe a numerical, univariate graph
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What strikes you as the most distinctive
difference among the distributions of exam scores
in classes A, B, C ?
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1. Center

• discuss where the middle of the data falls
• three types of central tendency
• mean, median, mode

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What strikes you as the most distinctive
difference among the distributions of scores in
classes D, E, F?
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2. Spread

• discuss how spread out the data is
• refers to the variability of the data
• Range, standard deviation, IQR

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What strikes you as the most distinctive
difference among the distributions of exam scores
in classes G, H, I ?
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3. Shape

• refers to the overall shape of the distribution
• symmetrical, uniform, skewed, or bimodal

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What strikes you as the most distinctive
difference among the distributions of exam scores
in class K ?
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4. Unusual occurrences

• outliers - value that lies away from the rest of
  the data
• gaps
• clusters
• anything else unusual

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5. In context

• You must write your answer in reference to the
  specifics in the problem, using correct
  statistical vocabulary and using complete
  sentences!

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More graphs for numerical data
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Stemplots (stem leaf plots)

• Used with univariate, numerical data
• Must have key so that we know how to read numbers
• Can split stems when you have long list of leaves
• Can have a comparative stemplot with two groups

Would a stemplot be a good graph for the number
of pieces of gun chewed per day by AP Stat
students? Why or why not?
Would a stemplot be a good graph for the number
of pairs of shoes owned by AP Stat students? Why
or why not?
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Example The following data are price per ounce
for various brands of dandruff shampoo at a local
grocery store. 0.32 0.21 0.29 0.54 0.17 0.28 0.36
0.23 Can you make a stemplot with this data?
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Example Tobacco use in G-rated Movies Total
tobacco exposure time (in seconds) for Disney
movies 223 176 548 37 158 51 299 37
11 165 74 9 2 6 23 206 9 Total tobacco exposure
time (in seconds) for other studios
movies 205 162 6 1 117 5 91 155 24 55 17 Make a
comparative stemplot.
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Histograms

• Used with numerical data
• Bars touch on histograms
• Two types
• Discrete
• Bars are centered over discrete values
• Continuous
• Bars cover a class (interval) of values
• For comparative histograms use two separate
  graphs with the same scale on the horizontal axis

Would a histogram be a good graph for the fastest
speed driven by AP Stat students? Why or why not?
Would a histogram be a good graph for the number
of pieces of gun chewed per day by AP Stat
students? Why or why not?
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Cumulative Relative Frequency Plot(Ogive)

• . . . is used to answer questions about
  percentiles.
• Percentiles are the percent of individuals that
  are at or below a certain value.
• Quartiles are located every 25 of the data. The
  first quartile (Q1) is the 25th percentile, while
  the third quartile (Q3) is the 75th percentile.
  What is the special name for Q2?
• Interquartile Range (IQR) is the range of the
  middle half (50) of the data.
• IQR Q3 Q1