Frequency tables &
Graphical techniques

SOC 221 • Lecture 2

Victoria Sass

Monday, June 24, 2024

Frequency tables

What do we do with data?

Apply some STATISTICS to answer interesting questions…

Descriptive statistics

Descriptive statistics
Procedures that help us organize
summarize, and describe the
distribution of a variable.

Distribution
The list of the entire set of observed
values of a variable, indicating all values
and how often those values occur.

  • Descriptive statistics allow us to summarize information in a distribution more efficiently
  • The type of descriptive statistics we can use to summarize a variable depends on how the variable is measured (i.e. its LEVEL OF MEASUREMENT)
Person # birth_year gender ethnicity race
1 1992 1 1 5
2 1993 2 0 5
3 1995 2 0 2
4 1980 2 0 3
5 1991 1 1 5
6 1975 4 1 1
7 1960 1 0 6
8 1952 1 0 5
9 2000 3 0 3
10 1990 1 1 2
11 1993 2 1 4
12 1992 3 0 4

Each column of our dataset contains the distribution of a different variable

Frequency tables: overview

Purpose of a frequency table

  • Frequency distribution table = descriptive tool to display the frequency of cases with various values/scores on the variable of interest.
  • Often present frequencies in multiple ways
    • raw frequencies
    • relative frequencies (percentages)
    • cumulative frequencies
    • cumulative percentages
    • etc.

Frequency table: step 1

Record raw data


Movie ratings (scale from 0 to 10 for 110 viewers)

>   [1]  5  8  3  6  5  8  3  7  9 10  6  8  7  4  6  5
>  [17]  8  7  4  6  5  3  6  1  8  6  8  5  6  6  8  9
>  [33]  6  9  6  5  9  9  6  5  8  5  8  4  8  6  7  4
>  [49]  8  6  8  7  6  8  5  7  6  9  6 10  4  8  6  8
>  [65]  6  5  4  7  8  5  7  4  5  5  9  6  6  7  5  3
>  [81] 10  3  5  7  9 10  6  7  4 10  6  9  8  7  8  5
>  [97]  7  4 10  6  7  4  2  3  8  5  4  7  7  7

Frequency table: step 2

Count the number of cases for each category


0
1 /
2 /
3 ///// /
4 ///// //// /
5 ///// ///// ///// //
6 ///// ///// ///// ///// ///
7 ///// ///// ///// //
8 ///// ///// ///// ////
9 ///// ////
10 ///// /

Frequency table: step 3

Place in raw frequency table

Table 1: Raw Frequency Distribution of Movie Ratings
Movie Rating
(score)		 Frequency
(f)
0 0
1 1
2 1
3 6
4 11
5 17
6 23
7 17
8 19
9 9
10 6
Total (N) 110

Descriptive title

First column: all possible values (sometimes labeled with a variable symbol, e.g. x )

Second column: number of times each value appears in the data

N = total number of cases in the data

Frequency table: step 4:

Add a percentage (relative frequency) column

Table 1: Raw Frequency Distribution of Movie Ratings
Movie Rating
(score)		 Frequency
(f)		 Percent
(%)
0 0 0
1 1 0.9
2 1 0.9
3 6 5.4
4 11 10
5 17 15.5
6 23 20.9
7 17 15.5
8 19 17.3
9 9 8.2
10 6 5.4
Total (N) 110 100

\[ Pct_x = \frac{f_x}{N} (100) \]

\[ Pct_0 = \frac{0}{110} (100) \]

\[ Pct_5 = \frac{17}{110} (100) \]

Percentages always add to 100

Frequency table: step 4.1:

Make comparisons across categories

Table 1: Raw Frequency Distribution of Movie Ratings
Movie Rating
(score)		 Frequency
(f)		 Percent
(%)
0 0 0
1 1 0.9
2 1 0.9
3 6 5.4
4 11 10
5 17 15.5
6 23 20.9
7 17 15.5
8 19 17.3
9 9 8.2
10 6 5.4
Total (N) 110 100

Comparing relative frequency of different values \[ Ratio = \frac{Pct_x}{Pct_y} = \frac{f_x}{f_y} \]

Ratio of 5s to 10s: \[ \frac{15.5}{5.4} = 2.87 \]

Frequency table: step 4.2:

Make comparisons across groups

Table 1: Frequency Distribution of Movie Ratings by Gender
(a) Females
Movie Rating
(score)		 Frequency
(f)		 Percent
(%)
0 0 0.0
1 0 0.0
2 0 0.0
3 1 2.0
4 3 6.0
5 5 10.0
6 9 18.0
7 10 20.0
8 12 24.0
9 6 12.0
10 4 8.0
Total (N) 50 100
(b) Males
Frequency
(f)		 Percent
(%)
0 0.0
1 1.7
1 1.7
5 8.3
8 13.3
12 20.0
14 23.3
7 11.7
7 11.7
3 5.0
2 3.3
60 100

Gender ratio of 10s: \[ \frac{8.0}{3.3} = 2.42 \]

Frequency table: step 5

Add a cumulative frequency column

Table 1: Raw Frequency Distribution of Movie Ratings
Movie Rating
(score)		 Frequency
(f)		 Percent
(%)
Cf
0 0 0 0
1 1 0.9 1
2 1 0.9 2
3 6 5.4 8
4 11 10 19
5 17 15.5 36
6 23 20.9 59
7 17 15.5 76
8 19 17.3 95
9 9 8.2 104
10 6 5.4 110
Total (N) 110 100

Cumulative frequency (Cf) =
Number of cases in the given category and lower categories

Cf in highest category = N

Frequency table: step 6

Add a cumulative percentage column

Table 1: Raw Frequency Distribution of Movie Ratings
Movie Rating
(score)		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
0 0 0 0 0
1 1 0.9 1 0.9
2 1 0.9 2 1.8
3 6 5.4 8 7.2
4 11 10 19 17.2
5 17 15.5 36 32.7
6 23 20.9 59 53.6
7 17 15.5 76 69.1
8 19 17.3 95 86.4
9 9 8.2 104 94.6
10 6 5.4 110 100
Total (N) 110 100

Cumulative
percentage (C%) =
Percentage of cases in the given category and lower categories

\[ C\%_x = \frac{Cf_x}{N} (100) \]

\[ C\%_4 = \frac{19}{110} (100) \]

C% in highest category = 100

Things to watch out for

Reverse ordering of categories

Frequency Distribution of Number of Statistics Classes Completed
Number of
courses		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
6 2
5 4
4 8
3 24
2 26
1 18
0 3
Total (N) 85

Things to watch out for

Reverse ordering of categories

Frequency Distribution of Number of Statistics Classes Completed
Number of
courses		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
6 2 2.35 85 100.00
5 4 4.71 83 97.65
4 8 9.41 79 92.94
3 24 28.24 71 83.53
2 26 30.59 47 55.29
1 18 21.18 21 24.71
0 3 3.53 3 3.53
Total (N) 85 100

Things to watch out for

Level of measurement makes some columns inappropriate

Frequency Distribution of College Majors

Major		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
Arts 61
Humanities 385
Social Science 425
Business 265
Physical Sciences 194
Other 10
Total (N) 1340

Level of measurement?

Things to watch out for

Level of measurement makes some columns inappropriate

Frequency Distribution of College Majors

Major		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
Arts 61 4.55
Humanities 385 28.73
Social Science 425 31.72
Business 265 19.78
Physical Sciences 194 14.48
Other 10 0.75
Total (N) 1340 100

Frequency or % “above” or “below” makes no sense without inherent ordering

Things to watch out for

What seems strange here?

Frequency Distribution of Satisfaction with SOCI 708

Response		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
Very Dissatisfied 7 9.33 7 9.33
Somewhat Dissatisfied 22 29.33 29 38.67
Somewhat Satisfied 20 26.67 49 65.33
Very Satisfied 8 10.67 57 76.0
No Answer 18 24 75 100.00
Total (N) 75 100

Things to watch out for

Missing information

Frequency Distribution of Satisfaction with SOCI 708

Response		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
Very Dissatisfied 7 9.33 7 9.33
Somewhat Dissatisfied 22 29.33 29 38.67
Somewhat Satisfied 20 26.67 49 65.33
Very Satisfied 8 10.67 57 76.0
No Answer 18 24 75 100.00
Total (N) 75 100


Question: Was a majority at least somewhat dissatisfied with the course (somewhat dissatisfied or worse)?

Things to watch out for

Missing information

Frequency Distribution of Satisfaction with SOCI 708

Response		 Frequency
(f)		 Percent
(%)
Cf
CPcnt
Very Dissatisfied 7 12.28 7 12.28
Somewhat Dissatisfied 22 38.60 29 50.88
Somewhat Satisfied 20 35.09 49 85.96
Very Satisfied 8 14.04 57 100
No Answer 18 24 75 100.00
Valid (N) 57 100


Question: Was a majority at least somewhat dissatisfied with the course (somewhat dissatisfied or worse)?

Best practice: Calculate all statistics based on information from those cases with valid (known) information on the variables of interest.

Break!

Graphical techniques


Graphical Techniques
Descriptive tools to
display the distribution
of variables, or the
association between
variables, using
pictures or figures.





  • Huge topic that people spend their lifetimes perfecting


Today’s goals

  • Brief intro to basic types
  • Basic guidelines for responsible graphing (given great potential to confuse or mislead)

5 rules for responsible graphs

  1. Choose the right kind of graph
    • Level of measurement
    • Bar charts, histograms, line charts etc.
  2. Clearly label the nature of the information
    • Title and axis labels are standard
  3. Focus on your perspective
    • Watch for angles, axis dimensions that skew results
  4. Keep it simple
    • Avoid extraneous, distracting information
    • Avoid the use of shapes that do not help convey information
  5. Be objective!

Rule 1: Choose the right kind of graph

Nominal & ordinal variables

Bar chart/graph: % or number of cases in each category

  • Summarizes the distribution (relative size of the categories)
  • Questions:
    • Biggest category?
    • Smallest category?
    • How much bigger is the biggest category from the 2nd biggest category?

Rule 1: Choose the right kind of graph

Nominal & ordinal variables

Bar chart/graph: % or number of cases in each category

Categories do not necessarily need to add up to 100%

  • This chart likely contains responses to several different questions/items.
    • Did you drink on Monday?
    • Did you drink on Tuesday?
    • Etc.
  • Level of measurement of the variables produced by these items?
    • Nominal
  • This type of variable is called a dichotomous (2 values) or dummy (0, 1) variable.

Rule 1: Choose the right kind of graph

Interval variables

Histogram: summarizes the number of cases with each value (or range) on a variable

  • Usually used for continuous variables (this is why the bars touch)
  • All bars add up to 100% of individuals

Histograms tell us about the SHAPE of a distribution

  • Center:
    • Most common value or values
    • Where the cases tend to be clustered/centered
  • Spread:
    • Is there a lot of diversity (cases spread out across the values of the variable)…
    • Or are most cases clustered within a few values
  • Symmetry vs. Skewness

Rule 1: Choose the right kind of graph

Interval variables

Histogram: summarizes the number of cases with each value (or range) on a variable

  • Usually used for continuous variables (this is why the bars touch)
  • All bars add up to 100% of individuals



  • What can you say about the shape of this distribution?

Rule 1: Choose the right kind of graph

Time-series data

  • Time always on the horizontal (x) axis
  • Value of the variable always on the vertical (y) axis.

5 rules for responsible graphs

  1. Choose the right kind of graph
    • Level of measurement
    • Bar charts, histograms, line charts etc.
  2. Clearly label the nature of the information
    • Title and axis labels are standard
  3. Focus on your perspective
    • Watch for angles, axis dimensions that skew results
  4. Keep it simple
    • Avoid extraneous, distracting information
    • Avoid the use of shapes that do not help convey information
  5. Be objective!

Rule 2: Clearly label the nature of the information

5 rules for responsible graphs

  1. Choose the right kind of graph
    • Level of measurement
    • Bar charts, histograms, line charts etc.
  2. Clearly label the nature of the information
    • Title and axis labels are standard
  3. Focus on your perspective
    • Watch for angles, axis dimensions that skew results
  4. Keep it simple
    • Avoid extraneous, distracting information
    • Avoid the use of shapes that do not help convey information
  5. Be objective!

Rule 3: Focus on your perspective

Rule 3: Focus on your perspective

5 rules for responsible graphs

  1. Choose the right kind of graph
    • Level of measurement
    • Bar charts, histograms, line charts etc.
  2. Clearly label the nature of the information
    • Title and axis labels are standard
  3. Focus on your perspective
    • Watch for angles, axis dimensions that skew results
  4. Keep it simple
    • Avoid extraneous, distracting information
    • Avoid the use of shapes that do not help convey information
  5. Be objective!

Rule 4: Keep it simple

5 rules for responsible graphs

  1. Choose the right kind of graph
    • Level of measurement
    • Bar charts, histograms, line charts etc.
  2. Clearly label the nature of the information
    • Title and axis labels are standard
  3. Focus on your perspective~~
    • Watch for angles, axis dimensions that skew results
  4. Keep it simple
    • Avoid extraneous, distracting information
    • Avoid the use of shapes that do not help convey information
  5. Be objective!

Rule 5: Be objective

Example of axis truncation

What does this chart imply about divorce rates?

Note the difference in the X and Y axis.

Rule 5: Be objective

No consistent y-axis

Graphic Presented by Congressman Jason Chaffetz (R-UT) During Meeting About Planned Parenthood Funding. September 29, 2015

Rule 5: Be objective

Purposefully misleading

Fin!