NOMINAL DATA: Summarizing Univariate Distributions

A. Central tendency: mode = the category with the largest frequency

Example: Rape: Relationship of offender to the victim, Kansas, 1988

RELATIONSHIP	f
ACQUAINTANCE	219
STRANGER	213
FRIEND	47
HUSBAND/COMMON-LAW HUSBAND	32
EX-BOYFRIEND	31
BOYFRIEND	10
EX-HUSBAND	15
FATHER/STEP FATHER	20
BROTHER	8
GRANDFATHER	2
UNCLE	1
OTHER FAMILY/IN-LAW	14
UNKNOWN	141
TOTAL	782

SOURCE: Caputi, J. & D. E. H. Russell "Femicide" Speaking the Unspeakable" MS Sept./Oct., 1990

Q: What is the average or typical relationship of a rapist to the victim?

A: Look at variable and decide what type it is: This is nominal data. Look at the question. It is asking for central tendency. Mode = acquaintance

*If two categories are both equally large, there will be 2 modes. There will be no central tendency if all categories have the same frequency.

ORDINAL DATA: Summarizing Univariate Distributions

A. Median: Central tendency

1. Put variable in order.

2. Number of median is:

a) If N is an odd number,

1. If N is an even number, the median(s)  is/are both and if they are different ordinal values.  For example, if the set of ordinal values is 3,3,6,7,8,8; then the median would be both 6 and 7 as there is no ordinal value 6.5 in the set.

Example: Nudity Tolerance Score: (ungrouped data)

	La Jolla	San Diego
	91	98
	90	98
	87	83
	86	83
N = 12	85	83	N = 11
median ->	83	82	<- median
median ->	81	82
	78	71
	77	42
	76	31
	76	30
	73

Q: What is the average score in La Jolla? In San Diego?

A: In La Jolla, the median is both 83 and 81. In San Diego, the median is 82.

Example: Past Soc201 final grades (grouped data, ordinal w/odd number of cases)

Grade	f	Cum f	Cases
A	16	16	1-16
B	18	34	17-34
C	21	55	35-55
D	23	78	56-78
F	13	91	79-91
N =	91

Q: What is the average grade for Soc201 classes?

A: The average grade for Soc201 is "C." Since N is odd,

(N + 1) / 2 = (91 + 1) / 2 = 46. The median is the ordinal category that contains case # 46 not 46!

Example: Persons who have known someone with AIDS (grouped data, ordinal w/even number of cases).

SES (socio-economic status)	f	cum f	cases
low	190	190	1-190
middle	221	411	191-411
high	33	444	412-444
N =	444

SOURCE: GSS73-91 SURVEY SUBSET

Q: What is average SES of persons who have known someone with AIDS?

A: Median SES "middle class"

444 / 2 = 222, 444+2 / 2 =223 the median falls between cases 222 & 223.

Interval Data or continuous Ordinal Data: Summarizing Univariate Distributions

A. Median: Central tendency

In some instances, such as income, housing costs, etc., a small minority of people skew the mean average (i.e. inflate or deflate). Additionally, some ordinal data is rendered in a numerical form (i.e. IQ scores, GPA's, etc.). In these cases, the median average is used. The following formula allows for a precise numerical answer. This formula is ordinarily used for grouped data with intervals greater than 1.

Median (Mdn)= LTL + {i[N(.50) - cf_below ]/ f }

where cf_below = cumulative frequency below interval containing the median.

f = the number of cases in the interval containing the median.

i = width of interval

N = total number in sample

LTL = lower true limit of the interval containing the median

Example: Ages (at last birthday)

	Md	Ages	Cum f
18-19	19.00	23	23
20-21	21.00	44	67
22-23	23.00	44	111
24-25	25.00	52	163
26-27	27.00	50	213
28-29	29.00	79	292
30-31	31.00	74	366
32-33	33.00	66	432
34-35	35.00	72	504
36-37	37.00	83	587
38-39	39.00	85	672
40-41	41.00	79	751
42-43	43.00	72	823
44-45	45.00	62	885
		885

• Find middle case. N(.50) = 885(.50) = 442.5
• Find interval that contains the middle case. cf_below is the cumulative frequency of the category immediately before the median category. cf _below = 432.
• Figure the proportion of the interval the median category is. [N(.50) - cf_below] / f = (442.5 - 432) / 72 = .146
• Multiply by i (i = real upper limit - real lower limit of each interval - must be a consistent interval) 35.999 - 34.000 = 2. 2(.146) =.292
• Add to real lower limit of median interval - 34.000 + .292 = 34.29 years.
  

• Q: What is the median average age?
  A: The median average age is 34.29 years.
  

B. Mean: Central tendency (abbreviated called x-bar)

1. ungrouped data: = x / N

Where: x = variable

N = total in sample

Example:

	weight
	80
N = 3	90
	60
	230 = x

Q: What is the average weight? There are 3 meanings of "average": nominal data = mode, ordinal data = median, interval data = mean (unless disproportionately skewed by upper or lower end of the range in which case median is used.)

A: = 230 / 3 = 76.67

2. grouped data: = fm / N

Where: f = frequency of a category

m = midpoint of a category

N = total number of cases in sample

# of children	f	m	fm
1 - 3	400	2	800
4 - 6	300	5	1500
7 - 9	200	8	1600
10 - 11	100	11	1100
	1000 = N		5000 = fm

Example:

Q: What is the average number of children?

A: 5000 / 1000 = 5

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