The t Distributions
The t distributions are used when we don’t have enough information to use
the z distribution.
We have to use the t distribution when we don’t know the population
standard deviation or when we compare 2 samples to one another
As the sample size gets bigger, the t distributions begin to merge with the z
distribution because we gain confidence as more and more participants are
added to our study
Estimating Population Standard Deviation from the Sample
Using sample ID to Estimate population SD
We don’t know the true value of the population standard deviation (σ).
Our best estimate of this population parameter comes from our sample
statistic (i.e., estimate population SD using sample SD)
Recall that any sample will be less variable than its parent population
Without correction for this difference, sample variance and SD would give us
an underestimate of population variability.
When calculating sample variance and standard deviation, use n-1 instead of
n to adjust for the bias in sample variability
å (X - m)
å (X - M)
Step 1: calculate the sample mean
Step 2: Use this sample mean in the corrected formula for the standard
Calculating Standard Error for the t statistic
In a z test, you compare your sample to a population with a known mean and
In actual research practice, you often compare two or more groups of scores
to each other, without any direct information about the populations.
Often, nothing is known about the populations that the samples are supposed
to come from.
The t statistic:
o When the variability of the population is unknown (s), we use the t-
statistic. o The t-statistic will let us substitute sample variance (or standard dev.)
for population variance.
o This means we use ‘s’ instead of ‘σ’
o Using the sample variance we can estimate the standard error of the
o Instead of: use or
o The estimated standard error (sM) is used as an estimate of the real
standard error (M ) when the unknown.
o It is computed from the sample variance or sample standard deviation
and provides an estimate of the standard distance between a sample
mean(m) and the Population mean (µ).
Using Standard error to calculate the t statistic
The t-statistic is used to test hypotheses about an unknown population mean
(μ) when the value of σ is unknown.
T statistic: indicates the distance of a sample mean from the population
mean in terms of the standard error
The formula for the t-statistic has the same structure as the z-score formula,
expect the t-statistic uses estimated standard erMor (s ) in the denominator.
Using the z test, you learned that when the population distribution follows a
normal curve, the shape of the distribution of means will also be a normal
curve. However, this changes when you do hypothesis testing with an
estimated population variance.
Because our estimate of σ is based on our sample (s ), and because from
sample to sample our estimate of σ will change…there is more variation in
our estimate of σ , and more variation in the t distribution than in a z
Like the normal distribution, the t-distributions are bell shaped, symmetrica
land have a mean of 0.
However, t-distributions generally have more variability than the normal
distribution (flatter and more spread out).
The t-distribution is really a “family” of distributions, as the exact shape
changes with the degrees of freedom. The single-sample t Test
single-sample test: a hypothesis test in which we compare data from
one sample to a population for which we know the mean but not the
The t Table and Degrees of Freedom
Degrees of freedom: the number of scores that are free to vary when
estimating a population parameter from a sample.
Just how much the t distribution differs from the normal curve depends
on the degree s of freedom.
The t distribution differs most from the normal curve when the degrees of
freedom are low (because the estimate of the population variance is
based on a very small sample).
In general, the greater the sample size (n), the larger the degrees of
freedom (n-1), and the better the t-distribution will approximate the
Degrees of freedom describe the number of scores in a sample that are
free to vary.
For each parameter you estimate, you lose one degree of freedom.
When we use the t statistic, because the known sample mean is used to
replace the unknown population mean in calculating the estimated
standard deviation, one degree of freedom is lost.
Thus, df= n-1.
This difference in shape across t-distributions (depending on degrees of
freedom) affects how extreme a score you need to reject the null
As in the z-test, to reject the null hypothesis using a single sample t-test,
your sample mean has to be in an extreme section of the distribution of
If the distribution has more of its means in the tails than a normal curve
would have, then the point where the rejection region begins has to be
further out on the comparison distribution.
Thus, it takes a slightly more extreme sample mean to get a significant
result when using a t distribution than when using a standard normal
As your sample size approaches infinity, the t distribution approaches the