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# Independent-measures t-test

Explanations > Social ResearchAnalysis > Independent-measures t-test

## Description

The t-test gives an indication of the separateness of two sets of measurements, and is thus used to check whether two sets of measures are essentially different (and usually that an experimental effect has been demonstrated).

The independent-measures t-test (or independent t-test) is used when measures from the two samples being compared do not come in matched pairs.

It is used when groups are independent and all people take only one test (typically a post-test). In design notation, this is:

 R X O R O

The requirement for Variance homogeneity test may be measured with Levene's test. Results for this can be given in SPSS along with the t-test results.

## Calculation

The value of t may be calculated using packages such as SPSS. The actual calculation for two groups is:

t = experimental effect / variability

= difference between group means /
standard error of difference between group means

=  Dg / SEg

Dg = AVERAGE(Xt) - AVERAGE(Xc)

where Xt are the measures in the treatment group and Xc are the measures in the control group. Note that any minus sign is removed, so that 't' remains positive.

SEg = SQRT( VAR(Xt)/nt + VAR(Xc)/nc)

where n is the number of people in the group and VAR(X) is the variance of X.

VAR(X) = SUM((X-AVERAGE(X))2)/(n-1)

A single group can also be compared with a measure, M, taken elsewhere.

t = (AVERAGE(X) - M) / (STDEV(X) / SQRT(n) )

Where STDEV(X) is the standard deviation of the group.

## Interpretation

The resultant t-value is then looked up in a t-test table to determine the probability that a significant difference between the two sets of measures exists and hence what can be claimed about the efficacy of the experimental treatment.

### Effect

The t-value can also be converted to an r-value to measure effect, which can be calculated as:

r = SQRT( t2 / (t2 + DF))

where DF is the degrees of freedom. In a t-test, DF = N1 + N2 - 2.

## Example

The results of two sets of measures are as follows:

 X1 X2 (X1-X1-bar)^2 (X1-X2-bar)^2 5.24 7.53 30.92 22.66 6.33 8.47 19.97 14.61 6.44 9.81 18.97 6.18 7.27 9.81 12.44 6.16 8.45 9.92 5.51 5.61 9.14 10.36 2.75 3.73 10.42 11.44 0.15 0.71 11.75 12.92 0.90 0.40 12.41 13.56 2.58 1.60 12.76 13.60 3.84 1.73 13.28 13.86 6.15 2.46 13.62 14.06 7.93 3.12 14.31 15.04 12.30 7.54 14.75 16.44 15.63 17.21 15.83 17.54 25.33 27.60 n 15 15 Sum: 165.37 121.32 df = n-1 14 14 Variance: 11.81 8.67 X1-bar X2-bar Std Error: 1.17 Mean: 10.80 12.29 t : 1.27

Looking up the t-value in the t-test table, with degrees of freedom = 15+15-2 = 28, the whole row is greater than t (0.32) so no significance can be claimed.

## Discussion

If the samples can be matched, such as in before-and-after measurement, then you should use the matched-pair t-test.

Matched-pair t-test

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