Looking at Data

1
Looking at Data

• Distributions

2
Definition

• A characteristic of a subject that can be
  measured is a variable.

3
Types of Variables

• Quantitative Takes on numerical values.
• Continuous variables take on many values on a
  finely-grained scale. Examples are temperature,
  pressure, or speed measurements.
• Discrete variables take on values that tend to be
  choppy, or integers. Examples are the number of
  children, number of DVDs owned, number of times
  married.
• Categorical No inherent numerical scale, but
  groups.
• Male/Female, Yes/No
• Married/Single/Divorced/Hoping

4
Types of Variables

• Categorical No inherent numerical scale.
• Sex Male/Female
• Opinion Yes/No
• Race Ethnicity White/Black/Hispanic/Asian/Other
  
• Marital Status Married/Single/Divorced/Hoping

5
The Distribution

• The pattern of variation in a variable is called
  its distribution.
• The distribution records the numerical values of
  the variable and how often each value occurs.

6
MM Activity

• Not only does the total number of MMs in every
  bag differ, but the number of MMs per color
  also varies greatly.

7
Distribution Features

• Center or typical value.
• Spread or variation.
• Extreme values or outliers.
• Distribution shape.
• Symmetric
• Skewed
• Number of peaks (modes).

8
Center

• An important feature of a distribution is what is
  the typical value in this data list. That is,
  what is the center of the distribution.
• In graphical displays like a dotplot, the center
  is found by noting the region or values where
  most of the dots are located in the display.
• What is the typical number of raisins in a box?

9
Spread

• This refers to the amount and nature of the
  spread or variation in the data values.
• We can measure the amount of spread by finding
  the difference between the biggest and smallest
  values in the data list. This is called the
  range, RangeMax Min.
• Can also measure spread by locating the center of
  the distribution and then noting how much above
  and below the center catches the vast majority of
  the data.

10
Extremes

• The extremes of a data list can often be very
  informative.
• They can help us identify outliers, values that
  are very different from the rest of the values in
  the list.
• The gap in this histogram helps us visually
  identify a very strange observation.

11
What accounts for outlier?
Binwidth 2
Binwidth 1
12
Symmetric Distributions
13
Number of Peaks (Modes)
14
Skewness

• Skewed left data has a distribution with a very
  long tail that extends to the left. Few, very
  small values.
• Skewed right data has a long tail extending to
  the right. Few, very large values.

15
Stemplot

• Number of home runs per season hit by Babe Ruth
  as a NY Yankee 54,59,35,41,46,25,47,60,54,46,49,
  46,41,34,22
• Stemplot Display for these data
• 2 25
• 3 45
• 4 1166679
• 5 449
• 6 0

16
Stemplot

• Number of home runs per season hit by Babe Ruth
  as a NY Yankee 54,59,35,41,46,25,47,60,54,46,49,4
  6,41,34,22

17
Stemplot

• Home runs for Roger Maris as a NY Yankee 8,
  13,14,16,23,26,28,33,39,61.
• Stemplot Display
• 0 8
• 1 346
• 2 368
• 3 39
• 4 ?------- Gap !
• 5
• 6 1 lt- ------- Outlier !

18
Splitting the Stems

• Percent Adults completed HS (USA States)
• 6 45667788
• 7 01224455555666677777899999
• 8 0000111122234457
• Poor display, data too concentrated. Need to
  spread out display by splitting the stems. Break
  each stem into leaves 0-4, and another for leaves
  5-9.

19
Splitting the Stems

• Better display
• 64
• 65667788
• 7012244
• 755555666677777899999
• 800001111222344
• 857
• New display clearly shows skewed left pattern.
• MS64, KY65, AR66, AK87, UT85, MN82, GA71,
  WI79, NDSD77.

20
Side by Side Stemplots

• A way to compare two distributions.
• Ruth Maris
• 08
• 1346
• 522368
• 54 339
• 9766611 4
• 944 5
• 0 61

21
Histograms

• Very useful when have lot of data.
• Step 1 in construction is divide data into
  groups.
• Step 2 count number of observations in each
  group.
• Step 3 draw histogram with bar heights
  proportional to the frequency of observations in
  each group.

22
Histogram Example

• Ages of diabetic patients 48, 41, 57, 83, 41,
  etc.
• Arrange values into a frequency table by groups.

23
Histogram
24
Histograms vs Bar Graphs

• The bars touch in a histogram for quantitative
  data, bars separated for a bar graph of
  categorical data.

25
Histogram

• 1996 States Dataset, built into StatCrunch
• Metro is of states residents living in metro
  areas.
• Produce histogram and observe features.

26
Homework

• 1.1, 1.3, 1.15, 1.17, 1.18, 1.21, 1.31