1

Picturing Distributions with Graphs

Stat 1510 Statistical Thinking & Concepts

14 Sept 2011

2

Examining the Distribution of Quantitative Data

 Observe overall pattern  Deviations from overall pattern  Shape of the data  Center of the data  Spread of the data (Variation)  Outliers

3

Shape of the Data  Symmetric

– bell shaped – other symmetric shapes

 Asymmetric –  right skewed –  left skewed

 Unimodal, bimodal

4

Symmetric Bell-Shaped

5

Symmetric Mound-Shaped

6

Symmetric Uniform

7

Asymmetric Skewed to the Left

8

Asymmetric Skewed to the Right

9

Color Density of SONY TV

10

Outliers

 Extreme values that fall outside the overall pattern – May occur naturally – May occur due to error in recording – May occur due to error in measuring – Observational unit may be fundamentally

different

11

Histograms

 For quantitative variables that take many values

 Divide the possible values into class intervals (we will only consider equal widths)

 Count how many observations fall in each interval (may change to percents)

 Draw picture representing distribution

12

Histograms: Class Intervals

  How many intervals? –  One rule is to calculate the square root of the

sample size, and round up.

  Size of intervals? –  Divide range of data (max-min) by number of

intervals desired, and round to convenient number

  Pick intervals so each observation can only fall in exactly one interval (no overlap)

13

Usefulness of Histograms

  To know the central value of the group   To know the extent of variation in the group   To estimate the percentage non-

conformance, if some specified values are available

  To see whether non-conformance is due to shift In mean or large variability

14

Case Study

Weight Data

Introductory Statistics class Spring, 1997

Virginia Commonwealth University

15

Weight Data

16

Weight Data: Frequency Table

sqrt(53) = 7.2, or 8 intervals; range (260-100=160) / 8 = 20 = class width

17

Weight Data: Histogram

100 120 140 160 180 200 220 240 260 280 Weight

* Left endpoint is included in the group, right endpoint is not.

Num

ber o

f stu

dent

s

18

19

20

Histogram of Soft Drink Weight

21

Histogram of Soft Drink Weight

22

Stemplots (Stem-and-Leaf Plots)

  For quantitative variables   Separate each observation into a stem (first

part of the number) and a leaf (the remaining part of the number)

  Write the stems in a vertical column; draw a vertical line to the right of the stems

  Write each leaf in the row to the right of its stem; order leaves if desired

23

Weight Data 1 2

24

Weight Data: Stemplot

(Stem & Leaf Plot)

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Key

20|3 means 203 pounds

Stems = 10’s Leaves = 1’s

192

2

152 2

5

135

25

Weight Data: Stemplot

(Stem & Leaf Plot)

10 0166 11 009 12 0034578 13 00359 14 08 15 00257 16 555 17 000255 18 000055567 19 245 20 3 21 025 22 0 23 24 25 26 0

Key

20|3 means 203 pounds

Stems = 10’s Leaves = 1’s

26

Extended Stem-and-Leaf Plots

If there are very few stems (when the data cover only a very small range of values), then we may want to create more stems by splitting the original stems.

27

Extended Stem-and-Leaf Plots

Example: if all of the data values were between 150 and 179, then we may choose to use the following stems:

15 15 16 16 17 17

Leaves 0-4 would go on each upper stem (first “15”), and leaves 5-9 would go on each lower stem (second “15”).

28

Time Plots   A time plot shows behavior over time.   Time is always on the horizontal axis, and the

variable being measured is on the vertical axis.   Look for an overall pattern (trend), and

deviations from this trend. Connecting the data points by lines may emphasize this trend.

  Look for patterns that repeat at known regular intervals (seasonal variations).

29

Class Make-up on First Day (Fall Semesters: 1985-1993)

30

Average Tuition (Public vs. Private)