• data analysis has several steps (in the order presented)

 
• Reduce the complexity of information

 
• e.g. in a counseling session, there is a volumn of data, but it is usually too complex to use for research or evaluation unless it is systematically organized

 
• There is too much information in a program evaluation to make sense unless the data are organized

 
• describe the sample 

 
• e.g. if you collect information on all the clients in an agency, you need some systematic way to describe their characteristics--race, gender, age, level of functioning, etc.

 
• if research question was descriptive, answer the research question

 
• the numbers used to describe the sample is usually calleddescriptive statistics

 
• estimate parameters and/or test hypotheses 

 
• both are probabilistic and involve calculation of error

 
• parameter error: confidence interval

 
• e.g. M=6.5 (±0.4)
• "estimated mean is 6.5, between 6.1 and 6.9 at the 95% level of confidence 
• In 95 out of 100 samples this size the mean will be between 6.1 and 6.9

 
• hypothesis error: p-value

 
• e.g. M₁=6.5, M₂=7.4 (t=2.68, p<0.05)
• You can get a difference of 0.9 between groups by chance alone in no more than 5 out of 100 samples with groups of this size

 
• The numbers associated with sampling error and hypothesis testing are usually called inferential statistics

 
• Most statistics social workers use in practice are descriptive

 
• to reduce the complexity of the information gathered:

 
• put in CASE x VARIABLE matrix (also called a dataset orspreadsheet)

 
• eg: hypthesize that intensive case management & job training will reduce the time mothers spend unemployed when compared to traditional casework

• data collection form:

 

Service (circle one)

(1) ICM 

(2) casework

Days unemployed ___ ___ ___ ___

Number of Children ___ ___ 

Number in Household ___ ___

Gender (1) Female

(2) Male

Race Identification (1) African-American

(2) Asian-PI

(3) Hispanic

(4) Native American

(5) Mixed ____ & ____

(6) Other __________________ 
 

Case No  (N)	Service (N)	Days Unemp (R)	# Children  (R)	# Household  (R)	Gender (N)	Race (N)
001	1	125	1	2	1	1
002	2	273	4	7	2	6
003	2	200	2	4	2	4
004	1	12	2	3	1	3

 
 

• Describing the sample is done w/ 

 
• language (qualitative)
• e.g. it was difficult for many of the subjects to get transportation to and from the training program

 
• statistics (quantitive)
• e.g. ICM reduced the days unemployed compared to the casework condition (M_ICM=121 days, M_C=148 days, t=3.75, df=267, p<0.05)

 
• images (qualitative-quantitative) 
• e.g. The ICM condition reduced unemployability by 18%

 

 
 
 
 
 
 
 

• qualitative description enables you to describe the richness & depth of experience, and to identify themes

 
• "thick description"
• "Mary S. had been off work for nearly three years. With multiple impairments such as mental illness, addiction, and illiteracy, it was unlikely she would ever get - and keep - a job.

 
• quantitative description enables you to identify central tendencies and variation with precision; saves space

 
• function of statistics: describing the sample, making inferences about the population

 

 
 
 

• DESCRIBING THE SAMPLE according to its variables in numerical terms

 
• can use actual numbers or percentages

 
• how described depends on the level of measurement of variables

 
• can do univariate, bivariate, and multivariate descriptions

 
• univariate =>1 variable
• What is the average number of unemployed days?

 
• bivariate => relationship between 2 variables
• What is the average number of unemployed days for women in ICM and for women in Casework?

 
• multivariate => effect of more than one variable on a dependent variable
• Considering gender, race, number in household and number of children, and intervention condition, which is the best predictor of days umemployed?

 
• Univariate description

 
• measures of central tendency 

 
• mean (ratio of the sum of an attribute to the number of cases being summed

 
• median (the number which divides the sample in half)

 
• mode (the most frequently occurring number)

 

 
 
 
 
 
 
 
 
 
 
 

• use mean when variable is interval or ratio AND when use of the mean does not present a distorted or skewed picture

 
• eg: 5 people in a group, and their ages are 20, 21, 22, 23, and 64
• mean age = 30
• median age = 22, a better measure of central tendency
• better use median when data are skewed 
• skew => mean is substantially different than the median
• "positive skew" => mean>>median
• "negative skew" => mean<<median
• in above example of 5 group members, data are positively skewed because M>MD

 
• measures of dispersion

 
• in above example, just using EITHER the mean age or the median age to describe the age would be deceiving: need an indicator of diversity, difference, spread, or variation in age

 
• range (the difference between the highest and lowest value of a variable)

 
• range = H-L+1
• 64-20+1 = 45 years
• "ranged from 20 to 64"

 
• range is meaningful as a number only for interval level variable, or for ordinal variable being treated as if it were interval (e.g. IQ, test scores, attitude measures, clinical evaluation tools, etc.)

 

 
 
 
 
 
 
 
 
 

• standard deviation

 
• s = average difference between individual scores and the mean

 
• s is not usually done by hand, except on a calculator

 
• s = 19.0 yrs

 
• variance 

 
• s² = the total amount of difference in the variable

 
• s² is not usually done by hand

 
• s² = 362.5

 
• other measures of dispersion for ordinal, categorical variables

 
• Bivariate Description: measures of relationship between two important variables 

 
• relationship between categorical variable(s) and an interval dependent variable => ANOVA (F-test)

 
• e.g. describe relationship between days unemployed and race

 
• M_W=95, M_A-A=120, M_H/L=124, M_O=98
• question: are these means significantly different? Use ANOVA to find out

 
• if the categorical variable has only two values (e.g. gender), we use a special case of ANOVA called the t-test, or means test

 
• e.g. describe relationship between days unemployed and gender

 
• M_M=85, M_W=120
• question: is 85 different than 120? Use t-test to find out

 
• linear relationship between two interval variables => regression coefficient (B)

 
• e.g. describe relationship between days unemployed andnumber of children

 
• M_UNEMPLOYED=100, M_CHILDREN=2.3
• question: is unemployment directly related to number of children? Use linear regression to find the best straight line (y = bx + c) which describes the relationship between these two variables, where b is the regression coefficient (slope of the line) and c is the constant (the value of y when x is zero, or where the line crosses the y-axis) (unemployment=b*children +constant; 
• e.g. unemployment = #kids*1.5 + 81 days, so we predict someone with 4 kids would be unemployed 81+4*1.5 or 87 days

 
• a special case of b is Pearson correlation coefficient [r = b(s_x/s_y)] which, unlike B, does not depend of the units of measurement
• e.g. r=0.25 between days unemployed and the number of children

 
• relationship between two categorical or nominal variable => chi-square (²)
• e.g. relationship between improvement and gender
• We put the data in a contingency table and use ² to suggest whether there is a relationship between between the two variable or whether they are independent of one another
• in above example, ² tests the (null) hypothesis that there is no relationship between gender and outcome, e.g.

Improved No Change Worse

Men 25 30 10

Women 40 30 5

Note that 25/65 (38%) of men improve and 40/75 (53%) of women improve, but is that difference significant? Use ² to find out (it is NOT significant: the critical value for df=2, p<0.05 is 5.99, and the value for this outcome is 3.79, not enough)
 

• Multivariate Description

 
• relationship between two or more categorical variables and an interval dependent variable => ANOVA (F-test)

 
• if there are two or more dependent variables, e.g. attitude about work fare before and after an education program, we use a more general form of ANOVA called multivariate analysis of variariance, or MANOVA

 
• relationship between two or more interval variables and an interval dependent variables => multiple regression (B)

 
• If the dependent variable is categorical, must not use simple linear regression, but logistic regression

 

• Beyond Description: Inference about the connection between sample and population

 
• level of confidence in your description

 
• 95% LOC as norm (p<0.05)

 
• ² = 6.02 is a description of the relationship between two nominal variables

 
• ² = 6.02, p<0.01 is an inference that you can only get a ² as large as 6.02 in this population by chance 1 out of every hundred times

 
• p<0.01 is called the alpha value, and it is set by a researcher, who must decide how much chance he or she is willing to live with, ie. how much confidence is needed

 

 
 
 

• interval of confidence is the range of values where the statistic may actually lie in the population for a given level of confidence

 
• to infer the mean: M_pop = M_samp ± (sd of mean) (LOC)