Basic Social Statistics-Lecture02B-Sociology, Study notes of Social Statistics and Data Analysis

Download Basic Social Statistics-Lecture02B-Sociology and more Study notes Social Statistics and Data Analysis in PDF only on Docsity!

Chapter 2

Describing Variables

2.4 Measures of Central Tendency

Central Tendency Measures

Measures of central tendency summarize the scores of a frequency distribution into a single number – the average or value most typical of all the scores.

How familiar are you with such “average values” as:

• College GPA
• Baseball batting averages
• Dow-Jones industry stock market index
• Unemployment rate

In statistics, three commonly used central tendency measures are the mode, median, and mean, which can be calculated for grouped and ungrouped data.

The Mode for Grouped Data

One type of grouped frequency distribution shows the lower and upper limits of collapsed ordered scores. In this case, the grouped data mode is the midpoint of the category with the largest number of responses (i.e., highest percentage). Calculate midpoint by summing the limits and dividing by 2.

Find the modal category for AGE20, an ordered discrete variable created by recoding AGE into four 20-year intervals:

Modal value = __________________

AGE Frequency Percent 20-39 695 35. 40-59 752 38. 60-79 437 22. 80-99 92 4. Total 1976 100.

The Mode For Grouped Data

Another type of grouped data table displays categories without collapsed values. Its mode is the score or label of the category with the most cases (i.e., largest percentage).

Find the modal category for REGION4, a nonordered discrete variable with K = 4 categories:

Frequency Percent Valid 1 NORTH EAST 2 MIDWEST 3 SOUTH 4 WEST Total

355 435 731 502 2023

17. 5
18. 5
19. 1
20. 8

Modal region = ______________________

The Median for Grouped Data

Find the median for an ordered variable grouped frequency distribution by examining the cumulative percentages of a table. Report label, score, or midpoint of the category whose cumulative percentage exceeds 50.0% (the 50th^ percentile).

SEXFREQ “About how often did you have sex during the last 12 months?”

Frequency Valid Percent

Cumulative Percent Valid 0 NOT AT ALL 1 ONCE OR TWICE 2 ONCE A MONTH 3 2-3 TIMES A MONTH 4 WEEKLY 5 2-3 PER WEEK 6 4+ PER WEEK Total

416 149 176 243 285 309 108 1686

**24.

100.**

**24.

100.**

Mdn = ________________________________________

Mean for Ungrouped Data

MEAN : the arithmetic average of a continuous variable’s scores

N

i 1

i

N

Y

Y

• Variable Y has N total observations (cases)
• The i th^ case is denoted subscript i
• The i th^ case’s score is Yi
• : add all the Yi scores, from i = 1st^ to N th
• Divide this sum of all the scores by N

Mean for Grouped Continuous Data

For continuous variables in a grouped frequency distribution, used this formula to calculate a weighted mean for the grouped data:

 



K

i 1

i i

N

(f Y ) Y

K is the number of categories

fi is the frequency (count) of cases in the i th^ category

Yi is the score of the i th^ category (or its midpoint)

To find the mean, first recode the original six GSS scores into an approximate number of times per year. (A year has 52 weeks and 12 months; for “3+ per week,” I assumed a value of “4 per week”)

SEXFREQ Frequency^ Valid Percent^ Cumulative Percent

0 NOT AT ALL = 0 1 ONCE OR TWICE = 2 2 ONCE A MONTH = 12 3 2-3 TIMES A MONTH = 36 4 WEEKLY = 52 5 2-3 PER WEEK = 156 6 3+ PER WEEK = 208 Total

416 149 176 243 285 309 108 1686

**24.

100.**

**24.

100.**

The 2008 GSS asked, “About how often did you have sex during the last 12 months?” Response categories allow SEXFREQ to be treated as a continuous variable.

Grouped Age Mean

For this table, which has the lower and upper limits of grouped scores, first calculate each interval’s midpoint. Then, apply the formula for the mean of grouped data. AGE Midpoint Frequency 20-39 29.5 695 40-59 49.5 752 60-79 69.5 437 80-99 89.5 92 Total 1976

 (^)   

K

i 1

i i N

(f Y) Y _____________________________________

Mean of a Dichotomy

You can also apply the grouped data formula to find the mean of a dichotomous variable (coded 0-1), which simplifies to:

N

f

N

(f )(0) (f )(1)

N

Σ(f Y ) Y

i i 0 1 1 

  

where ( f 0 / N) equals the proportion (p 0 ) of cases with Yi coded = 0 and ( f 1 / N) equals the proportion (p 1 ) of cases with Yi coded = 1

Thus, the mean of a dichotomy is simply the proportion of cases in one of the two categories.