Comparing Your Sample Mean and Variance to that of Some Normative Group
I'm a PhD student from the school of nursing, Queensland University of Technology,
Australia. I've really found your stats web pages helpful and wanted to pass this on to
you. I'm in the process of writing up my data analysis and have found I'm often using
your pages as a guide.
Also, I wonder if you could help me with this question? I'm wondering how you actually go about comparing your data to other published norms
etc. That is, using SPSS, how do I compare my sample in terms of instrument mean
scores and so on to other samples when I only have the published summary statistics
(means/SD etc) and don't have access to the actual data set?
Use a statistical package (SAS, SPSS, or whatever) to get the mean, standard
deviation, and sample size for your data.
If you have mean, standard deviation, and sample size for the normative group,
and are comfortable with the assumptions of the independent samples t test, you can
compare your sample mean to the normative mean by computing t (or by constructing a
confidence interval for the difference between means) by hand or using an online
program such as that at Graphpad.com. Suppose that you are using Wuensch’s Animal
Rights Survey. Wuensch (2002) reported M = 2.40, SD = .54, and N = 154. In your
sample M = 2.37, SD = .46, and N = 176 (actual data from research by Sharp et al.,
2006). You enter these stats at the Graphpad.com site, select the Welch test (a
separate variances tests, which should always be employed when sample sizes are not
equal), and click “Calculate now.” The results indicate that the difference between your
sample and the normative sample falls well short of statistical significance, t(302) =
0.53, p = .59, CI.95 = -.08, .14.
If you are interested in comparing the variance of your sample with that of the
normative sample, you can employ one of several tests of the null of homogeneous
variances.
If you have the normative mean but not the N for the normative sample, you can
conduct a one sample t test of the null hypothesis that your sample was randomly
drawn from a population with a mean equal to the normative mean or simply put a
confidence interval about your sample mean and see whether it contains the normative
mean or not. Suppose you are investigating correlates of scatophobia in a clinical
sample for which you suspect the patients may differ markedly from the normative
group. The test manual for the scatophobia scale you are using tells you that the
scores are standardized to mean 50, standard deviation 10. The normative sample is
described as having consisted of a large random sample of residents of Australia, New
Zealand, the United Kingdom, Canada, and the United States, but it does not give you
N.
You use SPSS Explore to get a confidence interval for mean scatophobia and
SPSS One-Sample T Test to conduct a test of the null that your sample was randomly
drawn from a population where the mean scatophobia score is 50.
Descriptives
Scatopho
Mean
95% Confidence
Interval for Mean
Statistic
54.472
52.609
Lower Bound
Upper Bound
Std. Error
.9283
56.335
As you can see above, the confidence interval for your sample does not include
the normative mean, so you can report that mean scatophobia in your sample is
significantly greater than that in the normative group.
One-Sample Statistics
N
Scatopho
Mean
53
Std. Deviation
54.472
Std. Error
Mean
6.7585
.9283
One-Sample Test
Test Value = 50
95% Confidence Interval
of the Difference
t
Scatopho
4.817
df
Sig. (2-tailed)
52
.000
Mean
Difference
4.4717
Lower
2.609
Upper
6.335
Here SPSS has given you more statistics that may chose to use to describe how
your sample differs from the normative group.
It is probably more important to report by how much your sample differs from the
normative group than whether the difference is significant or not. With very large
samples even trivial differences will be significant, and with small samples even large
differences will not be significant. Where the unit of measure is not intrinsically
meaningful, it should be helpful to report how large the difference is in standard
deviation units. For the example above your sample has a mean 4.472 points above
that of the normative group. How large is that in standard deviation units? For this
case, that depends on whether you use your sample standard deviation or the
normative standard deviation. In this case I think it makes more sense to use the
normative standard deviation, which is 10. Your sample has a mean .45 standard
deviations greater than that of the normative group. This statistic, by the way, is known
as Glass’ delta.
You should also note that your sample had a standard deviation lower than that
in the normative group. You could test the null that your sample was randomly drawn
from population where the standard deviation is the same as it is in the normative group
(10), but such hypotheses are rarely tested. Alternatively, you could report a confidence
interval for the sample standard deviation. Graphpad has an online program that will do
that for you. For our sample Graphpad calculates a 95% confidence interval that runs
from 5.67 to 8.36. Note that this does not include the normative value (10). The
scatophobia scores in our sample are significantly less variable than in the normative
group.
Reference
Sharp, H. W., Wuensch, K. L., Eppler, M. A., & Harju, B. L. (2006, April). Narcissism,
empathy, and attitudes towards animals. Presented at the Spring Conference of the
North Carolina Psychological Association and North Carolina Psychological
Foundation, Charlotte, NC.
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