SOCW 6311: Social Work Research in Practice II
Please note that this is a master level course so master level work. Please check the grammar, use APA format and you have to use the reading that I have provided to you. You must answer all the questions that I post. Thank you.
Week 4
Readings
• Dudley, J. R. (2014). Social work evaluation: Enhancing what we do. (2nd ed.) Chicago, IL: Lyceum Books.
o Chapter 9, “Is the Intervention Effective?” (pp. 226–236: Read from “Determining a Causal Relationship” to “Outcome Evaluations for Practice”)
• Plummer, S.-B., Makris, S., & Brocksen S. (Eds.). (2014b). Social work case studies: Concentration year. Baltimore, MD: Laureate International Universities Publishing. [Vital Source e- reader].
Read the following section:
o “Social Work Research:
Chi Square
” (pp. 63–65)
• Stocks, J. T. (2010). Statistics for social workers. In B. Thyer (Ed.), The handbook of social work research methods (2nd ed., pp. 75–118). Thousand Oaks, CA: Sage.
• Trochim, W. M. K. (2006). Internal validity. Retrieved fromhttp://www.socialresearchmethods.net/kb/intval.php
Be sure to click on all the links in the narrative.
• Document:
Week 4: A Short Course in Statistics Handout
(PDF)
© 2014 Laureate Education, Inc. Page 1 of 5
Week 4: A Short Course in Statistics Handout
This information was prepared to call your attention to some basic concepts underlying
statistical procedures and to illustrate what types of research questions can be
addressed by different statistical tests. You may not fully understand these tests without
further study. However, you are strongly encouraged to note distinctions related to type
of measurement used in gathering data and the choice of statistical tests. Feel free to
post questions in the “Contact the Instructor” section of the course.
Statistical symbols:
µ mu (population mean)
α alpha (degree of error acceptable for incorrectly rejecting the null hypothesis,
probability that results are unlikely to occur by chance)
≠ (not equal)
≥ (greater than or equal to)
≤ less than or equal to)
ᴦ (sample correlation)
ρ rho (population correlation)
t r (t score)
z (standard score based on standard deviation)
χ
2
Chi square (statistical test for variables that are not interval or ratio scale, (i.e.
nominal or ordinal))
p (probability that results are due to chance)
Descriptives:
Descriptives are statistical tests that summarize a data set.
They include calculations of measures of central tendency (mean, median, and mode),
and dispersion (e.g., standard deviation and range).
Note: The measures of central tendency depend on the measurement level of the
variable (nominal, ordinal, interval, or ratio). If you do not recall the definitions for these
levels of measurement, see
http://www.ats.ucla.edu/stat/mult_pkg/whatstat/nominal_ordinal_interval.htm
You can only calculate a mean and standard deviation for interval or ratio scale
variables.
For nominal or ordinal variables, you can examine the frequency of responses. For
example, you can calculate the percentage of participants who are male and female; or
the percentage of survey respondents who are in favor, against, or undecided.
Often nominal data is recorded with numbers, e.g. male=1, female=2. Sometimes
people are tempted to calculate a mean using these coding numbers. But that would be
© 2014 Laureate Education, Inc. Page 2 of 5
meaningless. Many questionnaires (even course evaluations) use a likert scale to
represent attitudes along a continuum (e.g. Strongly like … Strongly dislike). These too
are often assigned a number for data entry, e.g. 1–5. Suppose that most of the
responses were in the middle of a scale (3 on a scale of 1–5). A researcher could
observe that the mode is 3, but it would not be reasonable to say that the average
(mean) is 3 unless there were exact differences between 1 and 2, 2 and 3 etc. The
numbers on a scale such as this are ordered from low to high or high to low, but there is
no way to say that there is a quantifiably equal difference between each of the choices.
In other words, the responses are ordered, but not necessarily equal. Strongly agree is
not five times as large as strongly disagree. (See the textbook for differences between
ordinal and interval scale measures.)
Inferential Statistics:
Statistical tests for analysis of differences or relationships are Inferential,
allowing a researcher to infer relationships between variables.
All statistical tests have what are called assumptions. These are essentially rules that
indicate that the analysis is appropriate for the type of data. Two key types of
assumptions relate to whether the samples are random and the measurement levels.
Other assumptions have to do with whether the variables are normally distributed. The
determination of statistical significance is based on the assumption of the normal
distribution. A full course in statistics would be needed to explain this fully. The key point
for our purposes is that some statistical procedures require a normal distribution and
others do not.
Understanding Statistical Significance
Regardless of what statistical test you use to test hypotheses, you will be looking to see
whether the results are statistically significant. The statistic p is the probability that the
results of a study would occur simply by chance. Essentially, a p that is less than or
equal to a predetermined (α) alpha level (commonly .05) means that we can reject a null
hypothesis. A null hypothesis always states that there is no difference or no relationship
between the groups or variables. When we reject the null hypothesis, we conclude (but
don’t prove) that there is a difference or a relationship. This is what we generally want to
know.
Parametric Tests:
Parametric tests are tests that require variables to be measured at interval or ratio
scale and for the variables to be normally distributed.
© 2014 Laureate Education, Inc. Page 3 of 5
These tests compare the means between groups. That is why they require the data to
be at an interval or ratio scale. They make use of the standard deviation to determine
whether the results are likely to occur or very unlikely in a normal distribution. If they are
very unlikely to occur, then they are considered statistically significant. This means that
the results are unlikely to occur simply by chance.
The T test
Common uses:
To compare mean from a sample group to a known mean from a population
To compare the mean between two samples
o The research question for a t test comparing the mean scores between
two samples is: Is there a difference in scores between group 1 and group
2? The hypotheses tested would be:
H0: µgroup1 = µgroup2
H1: µgroup1 ≠ µgroup2
To compare pre- and post-test scores for one sample
o The research question for a t test comparing the mean scores for a
sample with pre and posttests is: Is there a difference in scores between
time 1 and time 2? The hypotheses tested would be :
H0: µpre = µpost
H1: µpre ≠ µpost
Example of the form for reporting results: The results of the test were not statistically
significant, t (57) = .282, p = .779, thus the null hypothesis is not rejected. There is not a
difference in between pre and post scores for participants in terms of a measure of
knowledge (for example).
An explanation: The t is a value calculated using means and standard deviations and a
relationship to a normal distribution. If you calculated the t using a formula, you would
compare the obtained t to a table of t values that is based on one less than the number
of participants (n-1). n-1 represents the degrees of freedom. The obtained t must be
greater than a critical value of t in order to be significant. For example, if statistical
analysis software calculated that p = .779, this result is much greater than .05, the usual
alpha-level which most researchers use to establish significance. In order for the t test
to be significant, it would need to have a p ≤ .05.
ANOVA (Analysis of variance)
Common uses: Similar to the t test. However, it can be used when there are more than
two groups.
The hypotheses would be
H0: µgroup1 = µgroup2 = µgroup3 = µgroup4
H1: The means are not all equal (some may be equal)
© 2014 Laureate Education, Inc. Page 4 of 5
Correlation
Common use: to examine whether two variables are related, that is, they vary together.
The calculation of a correlation coefficient (r or rho) is based on means and standard
deviations. This requires that both (or all) variables are measured at an interval or ratio
level.
The coefficient can range from -1 to +1. An r of 1 is a perfect correlation. A + means that
as one variable increases, so does the other. A – means that as one variable increases,
the other decreases.
The research question for correlation is: “Is there a relationship between variable 1 and
one or more other variables?”
The hypotheses for a Pearson correlation:
H0: ρ = 0 (there is no correlation)
H1: ρ ≠ 0 (there is a real correlation)
Non-parametric Tests
Nonparametric tests are tests that do not require variable to be measured at
interval or ratio scale and do not require the variables to be normally distributed.
Chi Square
Common uses: Chi square tests of independence and measures of association and
agreement for nominal and ordinal data.
The research question for a chi square test for independence is: Is there a relationship
between the independent variable and a dependent variable?
The hypotheses are:
H0 (The null hypothesis) There is no difference in the proportions in each category of
one variable between the groups (defined as categories of another variable).
Or:
The frequency distribution for variable 2 has the same proportions for both categories of
variable 1.
H1 (The alternative hypothesis) There is a difference in the proportions in each category
of one variable between the groups (defined as categories of another variable).
The calculations are based on comparing the observed frequency in each category to
what would be expected if the proportions were equal. (If the proportions between
observed and expected frequencies are equal, then there is no difference.)
© 2014 Laureate Education, Inc. Page 5 of 5
See the SOCW 6311: Week 4 Working With Data
Assignment
Handout to explore the
Crosstabs procedure for chi square analysis.
Other non-parametric tests:
Spearman rho: A correlation test for rank ordered (ordinal scale) variables.
• Document: Week 4 Handout: Chi-Square findings (PDF)
Week 4 Handout: Chi-Square Findings
The chi square test for independence is used to determine whether there is a relationship between
the two variables that are categorical in the level of measurement. In this case, the variables are:
employment level and treatment condition. It tests whether there is a difference between groups.
The research question for the study is: Is there a relationship between the independent variable,
treatment, and the dependent variable, employment level? In other words, is there a difference in
the number of participants who are not employed, employed part-time and employed full-time in
the program and the control group (i.e., waitlist group)?
The hypotheses are:
H0 (The null hypothesis): There is no difference in the proportions of individuals in the three
employment categories between the treatment group and the waitlist group. In other words, the
frequency distribution for variable 2 (employment) has the same proportions for both categories
of variable 1 (program participation).
** It is the null hypothesis that is actually tested by the statistic. A chi square statistic
that is found to be statistically significant, (e.g. p< .05) indicates that we can reject the
null hypothesis (understanding that there is less than a 5% chance that the relationship
between the variables is due to chance).
H1 (The alternative hypothesis): There is a difference in the proportions of individuals in the
three employment categories between the treatment group and the waitlist group.
** The alternative hypothesis states that there is a difference. It would allow us to say
that it appears that the treatment (voc rehab program) is effective in increasing the
employment status of participants.
Assume that the data has been collected to answer the above research question. Someone has
entered the data into SPSS. A chi-square test was conducted, and you were given the following
SPSS output data:
Assignment
Working With Data
Statistical analysis software is a valuable tool that helps researchers perform the complex calculations. However, to use such a tool effectively, the study must be well designed. The social worker must understand all the relationships involved in the study. He or she must understand the study’s purpose and select the most appropriate design. The social worker must correctly represent the relationship being examined and the variables involved. Finally, he or she must enter those variables correctly into the software package. This assignment will allow you to analyze in detail the decisions made in the “Social Work Research: Chi Square” case study and the relationship between study design and statistical analysis. Assume that the data has been entered into SPSS and you’ve been given the output of the chi-square results. (See Week 4 Handout: Chi-Square findings).
a 1-page paper of the following:
• An analysis of the relationship between study design and statistical analysis used in the case study that includes: ◦An explanation of why you think that the agency created a plan to evaluate the program
• An explanation of why the social work agency in the case study chose to use a chi-square statistic to evaluate whether there is a difference between those who participated in the program and those who did not (Hint: Think about the level of measurement of the variables)
• A description of the research design in terms of observations (O) and interventions (X) for each group.
• Interpret the chi-square output data. What do the data say about the program?
Statistics for Social
Workers
J. Timothy Stocks
tatrstrrsrefers to a branch ot mathematics dealing ‘”‘th the direct de
tion of sample or population characteristics and the an.ll)’5i• of popula·
lion characteri>tics b)’ inference from samples. It co•·ers J wide range of
content, including th~ collection, organization, and interpretJtion of
data. It is divided into two broad categoric>: de;cnptive >lathrics
and
inferential >lJt ost ics.
Descriptive statistics involves the CQnlputation of statistics or pnr.1meters to describe a
sample’ or a popu lation _~ All t he data arc available and used in <.omputntlon o f t hese
aggregate characteristics. T his may involve reports of central tendency or v.~r i al>il i ty of
single variables (univariate statistics). ll also may involve enumeration of the I’Ciation-
sh ips between or among two or moo·e variables’ (bivariate or multivariJte stot istics}.
Descriptiw statistics arc used 10 provide information about a large m.b> of data in a form
that ma)’ be easily understood. The defining characteristic of descriptive ;tJtistks b that
the product is a report, not .on inference.
Inferential statisti<> imolvc’ the construction of a probable description of the charac· Descriptive Statistics
Measures of Central Tendency Arir!Jmeric .\1ea11. The arithmetic mean usually is simply called the mca11. It also is called 75 76 PA11 f I • OuANTifAllVi AffkOAGHU: fouHo~;noM Of Ot.r”‘ CO ltf(TIO’J
~, =l:: X , 1
where 11 represents the popu I at ion mean, X represems an individual score, and rr is t he The formula for the sample mean is the same except t hat the mean is represented by – l:;X X= –. Following are t he numbers of class periods skipped by 20 seventh-graders d uring l:;X 219 • II 2 0
Mode. The mode is the most frequently appearing score. It really is not so much a measure quency distribution and determining which score has t he greatest fre-
TABLE 6 . 1 Truancy Scores Score
20 18 7
1 6
IS 4
1 3
1 2
II
10 6 5
4
3 2 frequ ency
2 3 2 0 I 0 0 Because 17 is the most frequently appearing number, the mode (or Unlike the mean or median, a distribution o f scores can have more ,llfedinrr. lf we take all the scores in a set of scores, place t hem in o rder There a.re 20 scores in the previous example. The median would be Measures of Variabi li ty de
information on how “spread out” scores in a d istribution are.
J If R
:.aJ de c .. …nu 6 • STAnsnu t<~~ Soc&AL Wouta~ 77
to the maximum ( highest) score. h is obtained by subtracting the 111ini murn score flom Let us compute th.- rang.- for the following dJt.l ~ct:
/1, 6, 10, 14, 18,22/.
‘T’he n1inimum i!) 2, and tht.” tnJximum is 22:
Range = 22 – 2 20.
Sum ofSquaus. The sum of squares is a measure of the total amount of variability in” set The formulas for sample and population sums ot squares are the same except for sam- SS = I(X ~tl’
Using the dJtJ set fo r t11e range, the sum of squnres would be computed as i n
‘ldble6.2.
V.~rinuce. Another name for variance i~ mean square. This is short for mean of squared ss n whc1e cr2 is the syn>bol for populn tion variance, SS is the symbol fo r sum of squares, and The variance for the example we used to compute TAOLE 6.2 Computing the Sum of Squares
X X m
2 tO
6 6
10 ]
l 18 >6
12 10
NOTE, !X~ 72; n- 6; ~ • 12; l:(X – p)’ ~ 780
(X – m)’
100
36
4 2 280 6 The sample variJnce is not an unbi.as.ed estin1a1o1 ss II – I CHA,Ut 6 • Sr”n~nn HJa SOCIAl wouus 77
to the maximum (highc;t) score. h is obtained by subtracting the minimum scoo·c from let us compute the rnnge for the following data set:
12. 6, 10, 14, 18.221 .
The minimum is 2. and the maximum is 22:
Range 22-2 = 20.
Sum of8qo~t~res. The ,um of squares;, a measure of the total amoun t o f variability in a set The formulas for <.omple and popul.llion sums of squares are the ~arne except tor S ss l.(X -X)’
Usi ng the data set for the range, t he su m of squares would be computed ns i n \~rta11u. Another name for variance is mean square. This is short for menn of 51JIUtred ss n where o ‘ is th e symbol foo· population v•o·ia.nc.e, SS is t he symbol fo o· Slim o f squares. a11d The •-..ria nee for the example we used to compute TABu 6.2 Computing the Sum of Squares
X X-m
2 – 10
6 -6
10 -2
14 +2
18 +6
22 +10
HOT£: r.x- 72: n; ti; p = 12: l:lX Ill’= 250.
(X- m)’
100 j(,
4 J&
tOO
280 6 The snmple variance is uot Jn \Ulbiased estimalor ss r =-. 78 PAll I • QuAiuu.ot.nvt A”MACH(S.:. FouHDAIIOif”i Of O.AIA CoLLfcnow
The n – 1 i> a correction fac tor for this tendency to undcre>tima te. I t is c.1 lled .1 280 6 – 1 Sumdard Deviatron. Although the variance is a measure of average variability associJtc’ Using the same .ct of numbers as before, the population standard deviation would be
cr -/46.67 = 6.83 .
and the sample st.mdard deviation would be
s J56 = 7.’18.
For a normally d istribured set of scores, n ppwximately 68% of all ;cores will be within Measures of Relationship One can use ,·eg,·cssion procedures to dcrivr the line that best fo ts the data. This line is Y,_. = 3.555 t 1.279X,
where Yis fi-equ ency o f Slope is the ch•ngc in Y for a unit increase in X. So, the slope of 11.279 meam that”” The equation does not give the actual value of Y (called the obt.tined or obserwd – r iQUIO 6.1 8
Frequency ol Stre 0 Punishment
~ c . Y P’td; – 3.555 + 1.279X .. 3 til 2 0
0 0 Stressors
example, if X were 3 , rhen we would predi<.t t hal Y would be - 3.555 + 1.279(3) ~ - 3.5
55
+ 3.837 ~ 0.282.
Tuu 6 . 3 frequency of Sue-ssors Pun1.shm~nt
3 0
4 4 }
s 3 7 ~
8 6
7 q 8
1() 9
T he regression li ne is the line that predicts Y >UCh t hat t he error E= Y Y..,.. ..
\\~1en X= 4, there arc two obL1ined ”alues of Y: I and 2. The Y,,…t = – 3.555 I 1.279( 4) = – 3.555 + S. l l6 ~ 1.56 1. E – 2 – 1.561 = +0.<139fnr Y=2 .
If we square each error difference score and sum the squares.
then we get a quantity called the enor sum of sq.ure;., which i;. SSI: L( Y – Y,..,.,)’.
T he regressi011 line io !he o ne line that give> the sm.11lcst va lue 80 P~oar 1 • QUAtHnAnvE A ,ROACHES: FouNOAHO~r~~$ of DAtA Conte I!Otf
The SSE is a measure of the lOla I variability of obtained score values around their pre- The total sum of squ.m:s (SS1) i$ a measure of the total variabilit)’ of the obtained SST = L(Y-Y)’.
The remaining sum of squa 1·cs is coiled the regression sum of S SSR L( v, …. – Y)’.
The SSR is a measure of the tot.d variabil ity of the predicted score values around the An important and interesting feature of the>e three sums of squares is that the sum of SST SSR- SSE.
This leads us to three o ther imponnnt stat istics: t he proportion of variance explJined Proportion of \Iarin nee Expluir~ctl. T ht I’VE is a measure of how good Lhc rcs,·cssion line SSR SST
There also is a computational equation for the PVE. which is
where
PVE – ( SSXY )’ SSXY is the “co variance” ~um of ;qua res: l.(X – X)( Y – Y ), The procedure fo r computing these sums of squares is outlined in Table 6.4. explained by stressors experienced ;,
( 4 6L5)1 3782.25 (48.1)(825) 3968.25 TABLE 6.4 Computation of r2 (PVE)
y Y – y (Y- Y)’ X X x (X – X)’ (X X)( Y Y) 4 -2 3 5.29 -lS 12 .25 +80S
4 -23 529 2 -15 6 .25 < 5.
75
5 – Ll 1.69 3 1.5 2.25 • 1.95
6 -ol 0 .09 < -o5 0.25 0 IS
7 +0./ 0.49 5 ·10.5 0.25 035
8 + II 2.89 6 ; 1.5 2 .25 • 2.55
7 TO.! 0.49 7 12.5 6 .25 11.75
9 +27 7.29 R t3.5 12.25 -19. 45
10 +3 I 13 69 9 “‘5 20.25 16. 65
NOTE: Y – 6.3; SSY – 48. l; X = 4.5; S5X = 82.5; S5XY • •6 l S
The PVEsometimes is en lied th~ coefticient of determination and is represented by the
teristics of a population b•sed on s.unple data. We compute statistics from .1 pJrtial;et of
the population data (a samplt) to estimate the population parameters. Thrse t
m:uics tO provide evidence for the exi
Measures of central tenden’)’ are individual numbers that typify the tot.tl set of ~cores.
The three most frequently used mca>urcs of centraltendenq are the arithmetic mean, the
mode, and the median.
the m-erage. It is computed b)’ adding up all of a set of scores and dwidmg by the number
of scores in the set. The algebraic representation of this is
1
number of scores being adde(l.
the variable lener with a bar above it:
II
I week: {1, 6,2,6, 15,2(),3,20, 17, 11, 15, 18,8,3, 17, 16, 14, 17,0, 101. Wecomputethe
mean by adding up the class periods missed and dh•iding by 20:
J.l = — = – = 10.9o.
of centrality as it is a measure of typicalness. It is found by o rganizing scores int o a fre-
quency. Table 6. 1 displays the truancy scores arranged in a frequency
distribution.
19
1
1
9
8
7
1
0
0
1
1
I
0
l
0
1
2
0
0
2
modal number) of class periods skipped is 17.
than one mode.
from least to greatest, and count in to the middle, then the score in the
middle is the median. This is easy enough if there is an odd number of
scores. However, if there is an even number of scores, then there is no
single score in the middle. In this case, t he two middle scores are
selected, and their average is the median.
the a”erage of the lOth and lith scores. We usc t he frequency table to
find these scores, which are 14 and J 5. T hus, the median is 14.5.
Whereas measures of central tendency are used to estimate a typical
score in a dimibution, measures of variability may be thought of ns a
way in which to measure departu re from typic<~lness. They pro"i
10
13
lhe maximum ~cor~.
of scores. Jts na me tells how to wmpute it. Smu ofsqunres is short (or sum ofsqumed dc1ti
til ion scores. It is represented by the S)’lnbol SS.
ple and populat•on mean symbob:
devintron score<. 1l1is is obtained by dividi ng the sum of squares by the number of scores
(11). It is a me,tsure of the average amount of variabilit y associated with each score in a set
of scores. The population variance fOI'mu la is
a2= -.
11 st,uJds for th e number of scores in the population.
sum of squares would be
4
36
100
(J –= 46.67.
of thf population variance. If we compute the vari
anccs for these samples using the SS/11 formula, then
the- san1ple vadn nccs wil1 average o ut smaller than
the population val’iance. For th is rc:~son, the sample
variance is computed differently froru the population
variance:
sl = – – .
the maximum score.
of score~. It> name tells how to compute it. Sum of 51Jo.arcs is short for ;um of squared dco•i-
atiou scores. It is reprewnt<>tl by the symlxll SS.
T.,b)e 6.2.
devontw11 scores. This os obtained by dividing the sum of squares by the number of ><.ores
(n). It is a measure of t he averoge ••m ount of var iability associated w ith each score in a set
of scores. T he popula tio n variance for m11ln is
¢ =- .
11 stands for the numbet of scores in the population.
sum of squar~s would be
4
cr2 =
~ 46.67.
o f’ t he population variance. Jf we com pute t he vari-
ances for these samples using th” SShr formu la, then
the sample variances will average out smaller than
thc population ••ariance. For this reJson, the sample
Vllriance is computed differe ntly from the population
variance:
n – J
degree• of freedon1. If
> =–
280 6
5 = 5.
age squared deviation from the mean. To get ” me
ll •tanrlard deviation of 1 he mean.
T.1ble 6.3 shows the relat iortship between number of >treSsors experien
rcfel’l’ed to as a regression line (or line of best ii 1 o r prediction I inc). Su ch a line bas been
.CJiculated for the example plot. It has a Y ime,·cept of – 3.555 t11id a slope of + 1.279. T his
gives us the prediction equation of
increase in stres.ors (X) of 1 will be accomp.ulicd by an increase in predicted frequency of
~orporal punishment (I’) of + 1.279 incidents per week. If the slope were a negati’e
number, then an increase in X would be accompanied by a pred ictcd decrease in Y.
score); rather, it giv~s a prediction of the value of Y for a certain value of X. Fo r
Cu,”na 6 • SrAliSnc
6 0
” 5 0 r:r
e …
c 4 ..
E
.r:
·;:
” Q.
0 1 2 3 4 5 6 7 8 9
Sttessors and Use of
Corporal Punishment
6 4
of p redictio n is minim ized. Error is d efined as the d ifference
between the predicted score and the obtaine
p redicted value of Y is
rhe error of prediction i~ E =I – 1.561 = -0.561 fu r Y = I, and
r~presented b)•
fo r SSt.
dicted values. There are two other ;un” of squares !hat are important to undcr>tanding
correlation and regri’SSion.
score values around the mean oft he obtained scores. The SST is represented by
mean of the obtained scores.
the SSR and SSE is equal to the SS1:
(I’VE) , the correlation coefficient, ond the standard error of estim ate.
p red icts obtained scores. The values of PV£ 1·ange fro m 0 ( no p red ictive value) to I ( pre-
diction with perfect accurJLy). The cqunt ion fo r PV£ is
J>vE – – ·
SSX • SSY’
SSX is t he sum of squares for vn rinble X: IlX – XJ’, and
SSYis the sum of squares for varinblc Y: 2:( Y – Y)’.
The proportion of v.triance in the freque ncy of corporal punishment thnl may be
l’VE = – = = 0 .953.
3 -33 10 .89 0 -4 5 20 .2 5 +1405
symbol r’.