1) appropriate for the measurement level of the variable

2) "rigidly defined rather than just approximated

3) based on all observations

4) simple & comprehensible

5) calculated with ease

6) expressed in algebraic terms

7) robust (little affected by fluctuations between samples)

possible other criteria

8) unique, rather than multivalue

9) generalizable to 2 or more variables

10) resistant to outliers

11) not overly affected by combining categories

12) defined even when variable has open ended categories

13) equal to actual data values, or at least in their metrics

if using samples

14) consistent for large samples

15) unbiased for small samples

16) efficient when compared to other possible estimators

levels of measurement

measurement - the process of assigning labels or values to observations

can divide variables into a couple of basic types

1) nominal - a series of unordered categories

its the lowest level

i.e. regions

we can examine the distribution

each category is listed with its corresponding frequency

total number of cases is N

can show the proportion of different categories

can also be represented as a percentage distribution

percentages are just proportions multiplied by 100

example

Immigration into US

by Last residence 1820-1988 in thousands

Ordinal scales

some nonnumeric variables have order to their categories

these are called ordinal variables

a prevalent type of ordinal data is ranked data

in dealing with ordinal data one can use the percentile

a percentile is a category of the variable below which a particular percentage of the observations fall

weak ordinal - where one can identify 'more than', 'less than' quality to 2 or more categories

strong ordinal - stronger ranking - but where exact quantitative difference between pairs of ranked items is unknown, i.e. a preference scale of preferred cities

metric scales - a variable that has a unit of measurement, ie. inches it answers the questions of how much or how many

2 major types: ratio and interval

ratio - is numeric with defined unit of measurement and a real zero point

because there is a zero point one can make ratio statements such as statement that one person is twice as tall as another

interval - defined unit of measurement but no real zero point

i.e. temperature

dichotomous variables - binary variables, very common in social science, it is nominal

categorization rules

1) mutually exclusive

2) mutually exhaustive

measures of centre

1) mode - simplest indicates which category is most common

main advantage is that its easy to compute

4 problems

a) may not be very descriptive

b) distribution may be bimodal or multimodal

c) can be overly affected by sampling variation

d) very sensitive to how categories are combined

when dealing with grouped metric data there is

a) crude mode = midpoint of most frequent interval

b) refined mode =

L = lower limit of the modal interval, w width of the class interval, f_mo = frequency of the modal interval, f _b the frequency of the interval below the modal interval, f_a = frequency of the interval above the modal interval

example

temp frequency

65-69 2

70-79 3

80-89 4

N = 9

mode = 80-89

crude mode 84.5

refined mode 81.5

example

median

for ordinal data = category of middle case

not meaningful to take averages of ordinal data

formula is median = category of the (N+1)/2th case

median for metric data

X_median = value of X for the (N+1)/2th ordered case for odd N

= [average of N/2th and (n/2)+1)]th ordered case for even N

advantages

1) not affected by extreme values

2) can sometimes be computed even if distribution is open on both ends

median for grouped metric data

rough median = midpoint of class containing middle case

exact median=

L = lower limit of the class containing the 50th percentile, f_median = freq of class containing 50th percentile, w = width of the interval of 50th percentile, C = cumulative freq below the 50th percentile class

mean for metric data

2 important properties

1) sum of deviations from the mean = 0

2) sum of + deviations = sum of - deviations

2 advantages

1) more stable than other measures

2) other important statistics can be derived using it

i.e variance and SD

problems

a) fractional values

b) cannot be computed if data is openended

c) strongly affected by extreme cases

mean for grouped data

the weighted mean

w_i= weight associated with ith case

weights compensate for the higher chances of selecting some cases than others

example is if pooled mean is of interest, values should be weighted by number of cases that made up each mean

mean for dichotomous data

where p is the proportion of successes or cases coded 1

these measures of central tendency tell us nothing about the variability in the data or the dispersion

one way to do this is compare the values with the mean value

the simplest way is to subtract the mean from each value to see if it is higher or lower

if you do this you get both + and - values

or the total squared variation about the mean

from this we can derive the variance and the standard deviation

variance is the sum of the squared deviations from the mean divided by N for the population and n-1 for a sample

remember that sample statistics are estimates of the population statistics

the sample uses n-1 because it has been shown that the use of N for a sample results in an underestimation of the population variance

a short cut formula for the sample variance is

a large standard deviation means a large variability in the data

variance can also be calculated for grouped data

where fⁱ=frequency of classes

M=grouped mean

empirical rule: if distribution is approximately then

mean + 1sd 68% of distribution

Mean +2sd 95% of distribution

mean +3sd 99.7% of distribution

example

9     90

4     40

6     60             s²=6(195)-961/6(6-1)=6.96

s=2.639

for b

x = 51.6

s²=696.6

s=26.39

coefficient of variation -

problem with variance and sd is that for the purpose of comparison, they are sensitive to the magnitude of the data

for example in the previous data the variance and sd of b was 10 times that of a

to compare a and b we need to standardize

coefficient of variation (cv) = s/x bar =std dev /mean of x

for a and b the cv is 2.639/5.155 = .51 or 26.39/51.6=.51

example application

measure the cv for different time periods for per capita income for pop

research question is: is the variation in income levels increasing or decreasing

for US 1880:.321, 1920:.291, 1940:.263, 1960:.176

variance is decreasing

previous measures don't provide any info about the shape of the distribution

another alternative is Pearson's coefficient of skewness that takes into account the position of the mean and median

a positive sign indicates a right (positive skew) a negative sign indicates a left of negative skew

as long as its within + or - 3 it can be considered moderate

there are alternative measures for skewness and kurtosis

like variance, skewness and kurtosis are an absolute quantity that is their values are affected by the magnitude of the data

to compare distributions we need relative measures

the bivariate mean

in geography the centre of an area may be of interest, can calculate the weighted bi-variate mean centre or the weighted centroid

standard distance

dispersion has it counterpart in bivariate descriptive statistics

because distances are deviations in the geographic sense, it is defined as the equivalent of a standard deviation

if you want to take possibility of an ellipse rather than a circle then we can calculate standard distance separately for X and Y

for weighted observations

this is far too tedious to do by hand so we would have to write a program to do it