The t-distribution is a theoretical distribution that is useful for estimating population means from small samples when the population standard deviation is unknown. It was discovered by William S. Gosset in 1908 and is characterized by a mean, median, and mode of 0. The document compares t-distribution to normal distribution and outlines applications of various t-tests.

1. T-distribution

2. What is T-distribution?
 T distribution is theoretical distribution that arises from
sampling from normal distribution
 Discovered by William S Gosset in 1908 and published under
penname ‘Student’ in Biometrika.
 Best for estimation of population mean when sample size is
small and population standard deviation is unknown

3. Specifics of T-distribution
PDF:
Mean = 0
Median = 0
Mode = 0
Variance = n / (n-2)
Skewness coefficient = 0
Kurtosis coefficient = 6 / (n-4)
2
1
2
1
2
,
2
1
1
)(
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













n
n
x
n
Bn
xf

4. T-Distribution vs
Normal Distribution
T distribution Normal distribution
More spread Less spread
More area in tails Less area in tails
Less area in centre More area in centre
Used for small n Used for large n
Population standard deviation
is unknown
Population standard deviation
is known

5. T-distribution Curve in
comparison with Normal Curve

6. Determination of T-table
critical values
Critical value tables are available for specified level of significance
and degrees of freedom.

7. Application of t-tests
 Test for single sample mean
 Independent samples t test
 Dependent sample t test
 Test for correlation coefficient

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