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# 2 Sample T vs Paired T Test

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Paired T Test is a hypothesis test used to determine if the means of 2 paired populations (same items are measured under different conditions) is same or not.

2 Sample T Test is a hypothesis test used to determine if the means of 2 independent populations (different items are measured under different conditions) is same or not.

An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Glory Gerald and Sanat Kumar.

Applause for all the respondents - Aritra Das Gupta, Irshad Patavekar, Glory Gerald, Sanat Kumar, Sudheer Chauhan, Anjali Parihar.

Also review the answer provided by Mr Venugopal R, Benchmark Six Sigma's in-house expert.

## Question

Q 304. Both 2 Sample T and Paired T tests are used to compare averages for 2 samples. What is the advantage of using a Paired T test? Explain with examples.

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• Solution

Two Sample T Tests are statistical tests that are used to compare the mean values of two independent samples/groups to determine if there is a significant difference between the means of 2 samples in reference. Two samples are considered to be independent if the selection of individuals/objects of one sample does not influence the selection of individuals/objects in the other sample in any way. The data from both samples should be normally distributed to apply the Two Sample T Test.

On the other hand, a Paired T Test is a statistical test that is used to compare the mean values of two related samples/groups to determine if there is a significant difference between the means of 2 samples in reference. A Paired T test is also called as  a Dependent T Test/Repeated Measures T Test. Here the groups can be related by being the same group of people, same item, or being subjected to the same conditions. Hence by using the same participants or item eliminates variation/individual differences that occur between the participants .Thus, Paired T tests are considered to be more powerful. This implies that we are more likely to detect a difference, if one does exist using a Paired T test over a Two Sample T Test. The differences between the values of the two related groups should be normally distributed to apply the Paired T Test.

Hypotheses of Paired T Test:

• The null hypothesis (H0) states that there is no significant difference between the means of two groups.
• The alternative hypothesis (H1) states that there is a significant difference between the means of two groups

Examples/Applications of Paired T Test :

• Performance of a group of students in a test conducted before and after a Training course.
• The before and after effect of a pharmaceutical treatment on the same group of participants.

• Body temperature using two different thermometers on the same group of participants.

• Body Weights of a group of participants before and after an exercise-training program.

• Body Weights of a group of participants before and after a diet counselling course

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In parametric statistics, when we have to compare means of two samples there are several test where 2 samples are dependent and independent

2 Sample T Test are used to compare the means of 2 sample which are not dependent on each other and have equal variance (determined by F test) which is normally distributed.

Example:

• If we want to compare performance of two team (sales performance)
• Comparing the runs scored by two different team
• Outcome of a drug testing on 2 independent groups

Wherein Paired Sample T test is performed to compare means of 2 samples which are dependent (paired) which is normally distributed. Two means could be :

• Comparison  pre and post-performance different times
• Comparison performance due to change in conditions  so on

Examples:

• Comparing performance of employee per and post refresher training
• Person’s health before and after a treatment

Benefits of using Paired T test

Paired T test is a powerful tool due to the below reasons:

1)      Since the sample used before and after are the same hence eliminated variation between the samples

• Same set of employee before and after training
• Same bike before and after servicing
• Same product before and after a marketing campaign

2)      Its best tool to measure the effectiveness of some factor on a sample, industries where it could be best in use are:

• Service Sector – measure performance of people before and after a training
• Sales – Effectiveness of awareness program/advertisement (pre and post sales value and volume)
• Pharma- effectiveness of medicine on patient’s health
• Automobile – Efficiency of automobile pre and post changing fuel efficiency

So, we can find its utility in all fields where ever comparative study is required on a paired

Sample

3)      Increase in the degree of freedom, in 2 sample T-Test df is “n1+n1-2 : as there are two different samples”, But in Paired it n-1 as the sample are same (decrease in df required high t power)

4)      Time and cost is minimized as the sample size remains the same (unlike 2 sample T test wherein 2 independent samples are collated

5)      The outcome of the two groups are co-related

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Benchmark Six Sigma Expert View by Venugopal R

When we have to compare the averages for two samples, it could be for different reasons:

1. To estimate whether two existing populations are different with respect to  their average values of the characteristics of interest.

Examples:

• To compare the average life span of bulbs produced by two different companies
• Average marks scored by male students vs that of female students

2. To estimate whether the effect of some change on a given population is significant or not.

Examples:

•  Performance of a group before training and after training
•  Average mileage of cars for one type of fuel vs another

From the above, we can see that for point-1, the two samples being compared can never be the same, since the reason for comparison is a difference based on the very nature of the sample itself. In such situations, we have to use 2-sample 't' test, and no ‘pairing' is possible.

For the point-2, we have a possibility of subjecting the same set of samples to the first treatment and then to the second treatment and compare the difference in performance for each same sample. In such situations, Paired ‘t’ test is the ideal comparative statistical tool to be used.

We may also come across some situations, where the paired sampling would not be practically possible. For example, let’s take the case of evaluating the average life of bulbs from the same company before and after doing a process improvement. Since the life testing of bulbs is a destructive test, the same samples will not be available for doing a paired 't' test and hence we have to use a different set of samples, and hence, only 2-sample 't' test.

Another example would be to compare the effect of two vaccines on a set of people. Once they are subject to vaccine-1, they would have developed immunity and we cannot subject the same set of people to vaccine-2, ruling out the possibility of a paired 't' test.

A paired ‘t’ test is recommended over 2-sample ‘t’ test whenever the situation permits, considering the advantages. Let me statistically illustrate certain advantages of paired test using the below example.

As part of a medical research study, the heart rates of 20 athletes were studied before and after subjecting them to a running program. Since heart rates of the same athletes were studied before and after the treatment, a paired test is possible. We will however, carryout the paired test and the unpaired 2-sample 't' test for the same sets of data and compare the results. The mean heart rate before the treatment was 74.5 and after treatment was 72.3.

The Minitab outputs for both the tests are given below:

From the above results, it can be seen from the p values that for the same set of data, the paired t test has shown significance, where as the 2-sample t test has not shown significance. Thus, the 2-sample t test for the same data exhibits higher ‘Type 2’ error.

Now, let us fix the required power of the test as 0.8 and determine the sample size requirements for both these tests, all data remaining same:

The above information are the outputs based on ‘Power & Sample size’. For both the type of tests, the sample size was determined based on a difference of 2, target power of 0.8 and standard deviation of 4.29. The paired test requires a sample of 39 whereas the 2-sample test requires a sample of 74.

Hence, the paired t test is preferable, whenever practically possible, from the sampling size requirement as well.

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Though Paired t test and 2 sample t test are used to compare 2 samples there is a fundamental difference as to when it should be used in a practical situation.

Similarly between Paired t test & 2 sample t test:-

1. Both are used when x is discreet and Y is continuous

2. Should be used when data is normal

3. Should be used only to compare 2 average of a sample

When should be Paired t test be used Vs 2 sample t test :-

2 sample t test should be used wen the two samples which has been selected for the statistical test are marginally independent and when that is not the case paired t test should be used

Example where paired t test should be used : -

1.   Medical - A new BP medicine is given to a set of 20 patient and the company wants to understand what is the impact of the medicine. Before & After results for the same set of patients should be taken and paired t test should be used whether the medicine has an impact on the BP.

2. Product - A company wants to check a new oil product and they use a test to conduct with bikes with and without the new product and compare there milage

3.  Service -A change in IVR has been introduced as a digital strategy and the before and after calls handled for a set of 100 agents is compared to check if there is a statistical significance of IVR on number of calls handled by agents

4. Medical -A new radiation technique is used for a set of 500 patients and test are compared for the same set of patients for the same set of patients

Example when t sample t test should be used : -

1. Service  - To compare the AHT/ NPS score between 2 different teams to check whether team leaders have an impact on overall AHT . This can be due to the way one team leader manages a team as compared to the other

2. Manufacturing - IF there is a difference  between the Mean time to repair for a CNC machine between one plant to another plant. This might be certain operator issue or other factors .

3. A pharmaceutical company -  A drug is produced to control type 1 diabetes and the test of the patients are compared with a set of patients who were not given the drugs. To check if there is a difference in Hyperglycaemia levels between two set of independent groups

4.Product - A heavy engineering company compares what is the voltage fluctuation between one transformer and other transformer which is a new one . This can help in understanding whether there is a reduction in fluctuation or there is no impact

5. HR - HR team has been receiving complaints that the average pay of a managers salary in 2 different metro cities with the same cost of living and employees with the same work experience. The team can take sample salary from a group of 20 managers from both the sites to compare the salary

Above are some examples when a paired t test is used Vs a 2 sample t test. While paired t test can be used when we have 2 sample and it is normally distributed . A 2 sample t test has to meet these as well as the criteria that there should be equal variance between 2 samples.

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2 sample t-test simply compares two different means from two samples. But the power of the paired t test is that it increases the sensitivity of the test without having to look at the series of other factors.

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2 Sample T test

2 sample t test method is a method used to compare the means of two population. We use 2 sample t test when the data of two samples are independent and it is also called independent samples t test.

Paired Sample T-Test

Paired sample t test is also called dependent sample t test and it is used to determine whether the mean difference between two sets is zero.

In a paired sample t test, each entity is measured twice, causing in pair of observation. Commonly uses of paired t test in case control studies or repeated -measured design for example if you want to check the effectiveness of company training program then you have to measure the performance of sample of employee before and after the program calculate the difference using a paired test.

Example: -

In clinical research , we usually compare the two group such as experimental and control and selection of statistical method is depend on type of data ,if data is continuous such as blood group and researcher may want to know the significant difference in mean value of two groups and if data is normally distributed then Two sample t test and paired t test are most commonly used method .

When comparing the two population. It is important to analysing that data samples from the population are two independent sample or one sample relate to pair (paired samples). Independent sample can be easily identified during the data generation. If one sample does not influence the selection of other individuals in any way in this case t sample 2 test should be applied .in other hand if one sample are coupled with some other particular observation in the other sample then called a paired .it is very common in before and after comparisons and in this situation we should use paired sample t test.

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These two tests are used to compare the means of two sets of observations but in paired T tests all the other external conditions are same.

e. g. If any medicine tested on same city people so external conditions are same for very one and people's responses to medicine are same but if the same medicine tested on two different cities people may react differently to medicine.

So variability due to people or objects in any observations comes out in this test.

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