ttest — t tests (mean-comparison tests) 11
The test for µ
x
= µ
y
for matched observations (also known as paired observations, correlated
pairs, or permanent components) is given by
t =
d
√
n
s
d
where d represents the mean of x
i
−y
i
and s
d
represents the standard deviation. The test statistic t
is distributed as Student’s t with n − 1 degrees of freedom.
You can also use ttest without the unpaired option in a regression setting because a paired
comparison includes the assumption of constant variance. The ttest with an unequal variance
assumption does not lend itself to an easy representation in regression settings and is not discussed
here. (x
j
− y
j
) = β
0
+
j
.
William Sealy Gosset (1876–1937) was born in Canterbury, England. He studied chemistry and
mathematics at Oxford and worked as a chemist with the brewers Guinness in Dublin. Gosset
became interested in statistical problems, which he discussed with Karl Pearson and later with
Fisher and Neyman. He published several important papers under the pseudonym “Student”, and
he lent that name to the t test he invented.
Stella Cunliffe (1917–2012) was an advocate for increased understanding of the role of human
nature in experiments and methodological rigor in social statistics. She was born in Battersea,
England. She was the first person from her local public girls’ school to attend college, obtaining
a bachelor of science from the London School of Economics. Her first job was with the Danish
Bacon Company during World War II, where she was in charge of bacon rations for London.
After the war, she moved to Germany and again helped to ration food, this time for refugees.
She then spent a long career in quality control at the Guinness Brewing Company. Cunliffe
observed that the weights of rejected casks skewed lighter. Noting that workers had to roll casks
that were too light or too heavy uphill to be remade, she had the scales moved to the top of
the hill. With workers able to roll rejected casks downhill, the weight of these casks began to
follow a normal distribution.
After 25 years at Guinness, Cunliffe joined the British Home Office, where she would go on to
become the first woman to serve as director of statistics. During her tenure at the Home Office,
she emphasized applying principles of experimental design she had learned at Guinness to the
study of such topics as birthrates, recidivism, and criminology. In 1975, she became the first
woman to serve as president of the Royal Statistical Society.
References
Acock, A. C. 2023. A Gentle Introduction to Stata. Rev. 6th ed. College Station, TX: Stata Press.
Boland, P. J. 2000. William Sealy Gosset—alias ‘Student’ 1876–1937. In Creators of Mathematics: The Irish Connection,
ed. K. Houston, 105–112. Dublin: University College Dublin Press.
Dixon, W. J., and F. J. Massey, Jr. 1983. Introduction to Statistical Analysis. 4th ed. New York: McGraw–Hill.
Earnest, A. 2017. Essentials of a Successful Biostatistical Collaboration. Boca Raton, FL: CRC Press.
Gosset, W. S. 1943. “Student’s” Collected Papers. London: Biometrika Office, University College.