Frequency Distributions

Chapter 3

Homework: 1, 2, 3, 12

Organizing Data

Describing distribution of variables enumeration: list raw data cumbersome to communicate

Frequency distributions organize ---> tables or graphs highlight important characteristics

• range, most frequent value ~

Distributions as Tables Frequency

# of times a value of variable occurs f = n

Tabular frequency distributions ordered list of all values of variable &

their frequencies except f = 0 logical order (usually descending) ~

Frequency Distribution

X f19 118 216 315 314 513 212 611 710 3 9 6 8 5 7 3 6 2 5 2

50

# of presentations to be able to recall 100%

8 9 7 816 710111614

121312131214 8 91512

18141412 81111 9 918

1511 7 9 5 6 8101111

101416 61115 91912 5

Enumeration

Grouped Frequency Distribution

Group by class intervals report f for intervals Lose information: exact values sacrifice detail for clarity

General rules each interval same width consecutive & do not overlap 6 - 20 intervals

Include intervals where f = 0 ~

Creating Grouped Frequency Distributions

1. Find range

highest - lowest score

2. Choose # of class intervals

3. Determine interval width range divided by # intervals round width to convenient # adjust # intervals if necessary ~

4. Determine lower limit of lowest interval should contain lowest data point and have convenient limits

5. Prepare list of limits, work bottom---> up highest interval must contain high

score

6. Count # of observations that occur at each interval f = n ~

Distributions as graphs

Summarizes data “A picture is worth ten thousand words”

Histograms & Frequency Polygrams Interval/ratio data grouped frequency distributions

Bar Graphs nominal or ordinal data frequency distributions ~

Bar Graphs

f

exam grades

2

6

10

14

18

A B C D F

Frequency Polygon

f

# of presentations

2

6

10

14

18

5 7 9 11 13 15 17 19 21

Histogram

f

# of presentations

2

6

10

14

18

5 7 9 11 13 15 17 19 21

Histograms

X-axis Class intervals of variables

Y-axis Frequencies represented as

vertical bars no spaces separating bars labels: lower or upper limit ~

Creating a Histograms

1. Start with grouped frequency dist.

2. Draw & label axes Y-axis 2/3 length of X-axis Y label: f, add tick marks & values X label: units for variable

e.g., pounds. seconds, inches, degrees add tick marks & value labels to X

evenly spaced, convenient round # ~

3. Y axis should intersect X at zero if not put in break to indicate same goes for Y axis

4. Draw vertical lines at edges of intervals single line, no spaces between

5. Bar height = frequency in interval

6. Provide explanatory notes e.g. where values on border belong clarity guiding principle ~

Frequency polygons

Contains same info as histogram Frequency represented as points

interval/ratio data, grouped f dist. Creating a frequency polygon

substitute single point for bar at midpoint of interval connect points with line ~

Relative Frequency

frequencies represented as percentage Large # of data points

Larger sample size ---> more intervals narrower bars ---> smoother curve ~

Relative Frequency

# of presentations

f

Shapes of distributions

Shapes of curves Unimodal distribution

single value is most frequent Bimodal

2 most frequently occurring values e.g. weight of all students

mode for female & mode for male distinct splitting of population does not imply no overlap ~

X

f Unimodaldistribution

X

f Bimodaldistribution

Symmetry of distributions

Symmetric if right side mirror-image of left unimodal or bimodal

Skewed - asymmetric tail on one side longer than other Positively skewed: right tail longer Negatively skewed: left tail longer Asymptotic: gradually approaches X

axis ~

Asymmetric Distributions

X

f

0 +2 +4-2-4

Positively skewed

X

f

0 +2 +4-2-4

Negatively skewed

The Normal Distribution

Characteristic of many distributions in nature bell-shaped

3 characteristics Unimodal symmetric asymptotic ~

Normal Distribution

f

Bar Graphs

Nominal and ordinal variables frequency represented by

separated bars Nominal variables

Histograms & Frequency polygons not appropriate

no intermediate values ~

Bar Graphs: Ordinal variables

Customarily use bar graphs Histograms & Frequency polygons

used occasionally Pro: indicates intervals not

necessarily equal Con: might imply no intermediate

values Decision: focus on clarity ~