Statistical Analysis for Education and Psychology Researchers

if t>1.98, decide on H 2 (Alternative hypothesis: μ>0)
The value 1.98 is obtained from a table of Student’s t distribution to give a significance
of 5 per cent (two-tailed test) for 113 (114–1) df. In this example the value of the t
statistic is 173.5, so you reject the null hypothesis and conclude that the sample mean is
not zero, and that in the population, the mean age of first class honours graduates is not
zero. Clearly this is a meaningless test in this example but it may be useful in some
applications.
The probability, P, associated with the t statistic is given in the column next to the t
value.
Sgn Rank is the signed rank statistic, computed as:

where is the positive rank of |xi|, where |xi| means the absolute
value of xi.
It is similar to the t statistic and is used to test the hypothesis that the median=0. This
test is only valid if the distribution is symmetric. In this example the (Wilcoxon) Signed
Rank test would not be valid. The sign test should be used.
M(Sign) is the sign test and is evaluated as M(Sign)=p−(n/2), where p is the number of values
greater than 0 and n is the number of non-zero values. Here the median sign test,
M(Sign)=114− 114/2=57. This statistic tests the null hypothesis that the population
median is zero. Associated with the statistic is the probability of obtaining a sign statistic
the same as or greater than the observed sample value. In this example we can reject the
null hypothesis and conclude that the population median is not zero.
Num^=0 is the number of values not equal to zero.
Sum
Wgts

is the sum of weights. Weights are assumed to be 1 unless defined otherwise.

Sum is the sum of scores (values).
Variance is a measure of variability about the mean. When the values are scattered widely about
the mean the variance is large. It is only ever zero if all the values are the same. It is
evaluated as:

where d is the specified df which depends on whether a population or a sample variance
is estimated.
CSS is the sum of squares corrected for the mean. It is calculated as the variance but not
divided by d (df).
Std Mean
is the standard error of the mean. This is calculated as

Quantiles

This section includes a selection of percentiles including the median and the upper and
lower quartiles, the interquartile range (Q 3 −Q1), the non-inclusive range, and the mode.
In this example the distribution of scores is unimodal. The mode unlike the mean and
sometimes the median represents an actual value in the distribution. The mean in contrast
represents a point on the distribution of values.

Extremes

The five lowest and highest values in the distribution are displayed along with
corresponding observation numbers.

Statistical analysis for education and psychology researchers 74