L.3 Two sample

Introduction

  • Welcome to Stats

  • Instructor: Ollie

  • Contact: 41151@uowmail.edu.au for questions after the lecture.

  • Unit coordinator: Mark, contact the 250 admin line for related questions.

Ollie's Background and Perspective

  • Background in mathematics and psychology.

  • First degree in mathematics with applied statistics.

  • Bachelor's honors PhD in psychology.

  • Aims to teach statistics in a way that is accessible and useful for psychology students.

  • Focus on practical application for research and employment.

  • Experience in a psychiatric hospital using statistics to evaluate patient progress.

  • Skills can be used beyond research, influencing career trajectories.

Lecture Slide Structure

  • Slides are written to be dense with information.

  • Designed to help students follow the lecture's narrative.

  • Encouragement for students to ask questions.

Experiments: Independent vs. Paired Designs

  • Dependent variable (outcome/effect) is on the vertical axis of a graph.

  • Independent variable (predictor/factor) is on the horizontal axis of a graph.

  • Experiment: Looking at how an independent variable influences an experimental group.

Independent Design

  • Different people in each condition.

  • Comparisons made between groups.

  • Also known as a between-groups design.

Paired Design

  • Same people experience each condition.

  • Also known as within-groups or within-individuals design, or repeated measures.

Example: Caffeine and Reaction Time

  • Guiding question: Does caffeine improve reaction time?

  • Independent design: Comparing reaction time between two groups (caffeine vs. no caffeine).

  • Paired design: Looking at reaction time before and after caffeine.

Independent Design: Random Allocation

  • Randomly allocate people to either an experimental group or a controlled.

  • Control group can receive a placebo or nothing.

  • Comparison: Caffeine versus no caffeine.

Paired Design: Repeated Measures

  • Same participants experience both conditions.

  • Measure reaction time with and without caffeine.

Terminology

  • Independent: Between groups, unpaired design, between subjects, cross-sectional, parallel groups.

  • Paired: Within subjects, within groups, repeated measures, longitudinal design (over a longer period).

Applying Designs to PhD and First-Year Students

  • Paired Design: Measures reaction times of PhD and first-year students, with repeated measures.

  • Independent Design: Comparing reaction time of PhDs with no caffeine versus first-year students with caffeine.

Features of Experiments

  • Participant assignment.

  • Type of comparisons.

  • Effect of individual differences.

  • Risk of order effects.

  • Statistical power.

  • Number of participants needed.

  • Confounding variables.

  • Cases of best used when.

Participant Assignment

  • Independent design: Different people in different groups.

  • Paired design: Same people in both conditions.

Type of Comparisons

  • Independent: Between groups.

  • Paired: Within the same group.

Effect of Individual Differences

  • Independent: High impact of individual variation, natural variations can provide added noise.

  • Paired: Lower impact, individual serves as their own control.

Risk of Order Effects

  • Independent: No risk.

  • Paired: Risk of order effects, e.g., caffeine still in blood.

Statistical Power

  • More data is good.

  • Independent: Lower statistical power.

  • Paired: Higher statistical power because differences between participants can be made larger when repeatedly measuring people.

Number of Participants Needed

  • More samples generally better.

  • Fewer participants needed for repeated measures design, but there is a minimum.

  • Avoid very small subgroups (e.g., two males in a class).

Confounding Variables

  • Variable that has an unintended effect on the dependent variable.

  • Independent: Differences between groups in intelligence, age, or central nervous system activity can exacerbate the effects.

  • Paired: Order effects can play a role.

Cases of Best Used When

  • Independent: Large diverse sample, no risk of order effects.

  • Paired: Participant variability needs to be controlled, fewer participants are available.

T-Tests

  • Invented to brew beer better.

  • Used to compare two things to see if they are meaningfully different.

  • Execution depends on whether it is an independent or paired design.

Independent T-Test

  • Comparing the mean reaction time between two independent groups.

Paired T-Test

  • Comparing the average reaction time within individuals across two conditions.

Choosing the Right Test

  • Depends on the hypothesis and practical factors like size, feasibility, and common sense.

  • Independent t-tests: Comparing between two separate groups.

    • E.g., People who consume caffeine will have a faster reaction time than those who do not.

  • Paired t-tests: Comparing the same people before and after.

    • E.g., Individuals will have a faster reaction time after consuming caffeine compared to their own baseline.

  • Hypothesis is a guess, or an if and then statement.

Research Question Example

  • Does caffeine improve reaction time more for first-year uni students or PhD students?

Steps

  • Identify the independent and dependent variables.

  • Determine the type of comparison.

Example 1: Independent Variable - Groups

  • Group 1: First-year uni students.

  • Group 2: PhD students.

  • Dependent variable: Reaction time.

  • Comparison between two groups.

  • Each participant belongs to only one group.

  • No within-individual comparison.

Example 2: Independent Variable - Condition

  • Condition 1: Reaction time of all students before having caffeine.

  • Condition 2: Reaction time of all students after having caffeine.

  • Reaction time is being compared within individuals, which results in a paired design.

Null Hypothesis

  • States that there is no effect or difference between the variables.

  • Default assumption that any observed differences are due to random chance.

  • Expressed as H0H_0

  • We want to determine if the data observed comes from random chance or from the experiment and is represented by statistical tests attached to some form of distribution (e.g. t distribution, z distribution, or f distribution).

T Distribution

  • Histogram where the middle shows the most common result (zero).

  • Comparing the test statistic to the distribution.

  • Can be based on p-values or critical values.

Directionality

  • Some hypotheses have a direction, others are non-directional.

  • Non-directional: Caffeine has any effect on reaction time.

  • Directional: Reaction time will be faster after having caffeine.

  • Non-directional tests split the p=0.05p = 0.05 value across the two ends of the distribution.

  • One-tailed tests refer to one side of the distribution.

Paired Samples T-Test

Research Question

  • Does caffeine improve reaction time?

Null Hypothesis

  • Caffeine has no effect on reaction time; no significant difference in reaction times before and after having caffeine.

Alternative Hypothesis

  • Caffeine affects reaction time; there will be a significant difference in reaction times before and after.

Directional Hypothesis

  • Caffeine improves reaction time; times after consumption faster than before.

Design

  • Everyone experiences the same condition. Repeated measures. Reaction time is measured before and after.

Data

  • Participants A through R.

  • Reaction time before and after measured in milliseconds.

Steps (Manual).

  • Calculate the difference in reaction times under each condition for each participant.

  • Calculate the average and standard deviation of all those difference values.

    • Average=sum of valuesThe number of valuesAverage = \frac{sum \ of \ values}{The \ number \ of \ values}

  • Calculate degrees of freedom.

    • df=n−1df = n - 1

  • Calculate the test statistic.

    • t=Average of differencesStandard DeviationNt = \frac{Average \ of \ differences}{\frac{Standard \ Deviation}{\sqrt{N}}}

    • That simplifies to: t=Average∗nStandard Deviationt = Average * \frac{\sqrt{n}}{Standard \ Deviation}

Directional Hypothesis (One-tailed Test).

  • Caffeine will reduce reaction time (reaction times should be significantly faster).

  • Undertake a one-tailed test to see if there's any difference.

  • Work out the odds of getting that t-value with the relevant number of degrees of freedom.

  • Probability of randomly obtaining those results needs to be really small to suggest caffeine does actually alter the results.

Nondirectional Hypothesis (Two-tailed Test).

  • Two tailed, don't care what direction results are going in (faster or slower), just interested to know what change has been made. Can test using all collected data.

Interpreting the Results.

  • P value around 1 in 12 million suggests that the tested T variable has a very slim chance of randomly influencing someone if caffeine is given to them. This is also known as an alpha point.

  • p < 0.05 so reject null hypothesis in favor of alternative hypothesis.

    • Supported. The data supports the hypothesis that it did cause reaction time to reduce.

  • However the previous results don't tell us what the direction is.

    • To know more need to use a One-Tailed Test hair test statistic, can calculate in a calculator around one in twenty five million. This again also enables you to reject a null hypothesis that caffeine does not improve reaction time.
      How to Write Up in APA.

  • Non-Directional Hypothesis Test

    • t(df) = value, p < 0.05

  • Directional Hypothesis

    • t(df) = value, p < 0.05

Do not be inconsistent with the number of decimal places in results. Recommendation. Stick to just two decimal places.

Independent Samples T-Test

Example

  • The example used here is about testing whether caffeine affects reaction time differently when examining a group of PHD students vs First Years.

Experimental Design

  • It involves measuring the reaction time of a group of PHD students after giving them nothing to ingest, and doing the same with a group of first years after giving them caffeine.

Outcome

  • There will be a significant difference between those from different experimental groups.

  • For an overall result it has to be hypothesized that giving a group with caffeine gives then faster reaction times.

Independent Sample

  • In this example all participants will only experience one condition. The variables under inspection are independent.

Setup Simulations:

Reaction time can be different between experimental groups. The reaction rate has two main components: Group 1 and Group 2. Group 1 average result= 254.00ms standard deviation: 21.8ms. Group 2 average results: 239ms; 33 standard deviation. Each group tested includes 9 people each.

Calculations:

  • Calculation the independent T-test statistic: Take group one mean, and minus group two mean. Divide by the square root of both the group 1 standard deviation squared / by its sample size and group twos equivalent result.

    • This can be reverse, the impact on the results will be that If it comes up negative, go for the abstract. If the P h result is higher: the value will need to show that group reaction time is higher vs first years. Swap around data results and analyse again. It's not actually going to affect test statistic.

  • Calculating the degrees of freedom: Degree of freedom group -1 + degree of freedom group 2 -1.

    • This means the total number for PHD + FIRST YEAR number of results -1 is 16. Since there's two sides to a hypothesis, so therefore two differing versions will test it.

Assumptions

* T- tests relies upon many assumptions. So if assumption criteria is not obeyed well then the test itself won't work. This does not apply to every exception. One is always the same, but we'll get back to it. The main important assumption being made, that the observations remain independent, not affecting each others results.
    *   Law Of Space.
    *   Each one should come from a totally seperate individual

There needs to be normality to the underlying distribution. What degree this is affected depends on test type being run. A test is said to be on normal dependent, but this really needs to be across the dependant variable within EACH group.

  • Homogeneity of variance this tends not to be a paired test/ test. What it really means in practical terms is, the spread (variance) of the data around is roughly equal across all the results.

  • Scale of measure: Needs to be measurable on interval on ratio scale. The values needs to be scale. An example would be

Introduction

  • Welcome to Stats. This course aims to provide a foundational understanding of statistical concepts and their applications.

  • Instructor: Ollie

  • Contact: 41151@uowmail.edu.au for questions after the lecture. Students are encouraged to reach out with any questions or clarifications needed on the lecture material.

  • Unit coordinator: Mark, contact the 250 admin line for related questions. For administrative queries or issues related to the unit, students should contact the unit coordinator through the provided admin line.

Ollie's Background and Perspective

  • Background in mathematics and psychology. Ollie's interdisciplinary background allows for a unique approach to teaching statistics, bridging the gap between theoretical concepts and practical applications.

  • First degree in mathematics with applied statistics. This provides a strong quantitative foundation for understanding statistical methods.

  • Bachelor's honors PhD in psychology. The PhD in psychology provides insights into the application of statistics in behavioral research.

  • Aims to teach statistics in a way that is accessible and useful for psychology students. The focus is on making statistics relevant and understandable for students in psychology.

  • Focus on practical application for research and employment. Emphasis is placed on how statistical skills can be applied in research settings and in various employment opportunities.

  • Experience in a psychiatric hospital using statistics to evaluate patient progress. Practical experience in using statistical methods to assess and monitor patient progress in a psychiatric setting.

  • Skills can be used beyond research, influencing career trajectories. Statistical skills are valuable beyond academic research and can significantly influence career paths in various fields.

Lecture Slide Structure

  • Slides are written to be dense with information. The slides are designed to be comprehensive, containing detailed information to support in-depth understanding.

  • Designed to help students follow the lecture's narrative. The structure of the slides is intended to guide students through the logical flow of the lecture.

  • Encouragement for students to ask questions. Students are encouraged to actively engage with the material and seek clarification on any unclear concepts.

Experiments: Independent vs. Paired Designs

  • Dependent variable (outcome/effect) is on the vertical axis of a graph. The dependent variable represents the outcome or effect being measured in the experiment and is typically plotted on the vertical axis of a graph.

  • Independent variable (predictor/factor) is on the horizontal axis of a graph. The independent variable is the factor being manipulated or controlled by the experimenter and is plotted on the horizontal axis of a graph.

  • Experiment: Looking at how an independent variable influences an experimental group. Experiments are conducted to examine the impact of the independent variable on the experimental group.

Independent Design

  • Different people in each condition. In an independent design, different participants are assigned to different experimental conditions.

  • Comparisons made between groups. Comparisons are made between the groups exposed to different conditions to determine the effect of the independent variable.

  • Also known as a between-groups design. Independent design is also referred to as a between-groups design because comparisons are made between different groups of participants.

Paired Design

  • Same people experience each condition. In a paired design, the same participants are exposed to all experimental conditions.

  • Also known as within-groups or within-individuals design, or repeated measures. Paired design is also known as within-groups, within-individuals, or repeated measures design because measurements are repeated on the same participants.

Example: Caffeine and Reaction Time

  • Guiding question: Does caffeine improve reaction time? This example explores whether caffeine has a positive impact on reaction time.

  • Independent design: Comparing reaction time between two groups (caffeine vs. no caffeine). An independent design would involve comparing the reaction times of two separate groups: one receiving caffeine and one receiving a placebo or no caffeine.

  • Paired design: Looking at reaction time before and after caffeine. A paired design would involve measuring the reaction time of the same individuals before and after caffeine consumption.

Independent Design: Random Allocation

  • Randomly allocate people to either an experimental group or a controlled. Participants are randomly assigned to either the experimental group (receiving the treatment) or the control group (not receiving the treatment).

  • Control group can receive a placebo or nothing. The control group may receive a placebo (an inactive substance) or no treatment at all.

  • Comparison: Caffeine versus no caffeine. The comparison is between the group receiving caffeine and the group not receiving caffeine to determine the effect of caffeine on reaction time.

Paired Design: Repeated Measures

  • Same participants experience both conditions. All participants undergo both experimental conditions (e.g., with and without caffeine).

  • Measure reaction time with and without caffeine. Reaction time is measured for each participant under both conditions to assess the effect of caffeine.

Terminology

  • Independent: Between groups, unpaired design, between subjects, cross-sectional, parallel groups. These terms are used interchangeably to describe designs where different groups of participants are compared.

  • Paired: Within subjects, within groups, repeated measures, longitudinal design (over a longer period). These terms refer to designs where the same participants are measured under different conditions or over time.

Applying Designs to PhD and First-Year Students

  • Paired Design: Measures reaction times of PhD and first-year students, with repeated measures. This involves measuring the reaction times of both PhD and first-year students under multiple conditions (e.g., before and after caffeine).

  • Independent Design: Comparing reaction time of PhDs with no caffeine versus first-year students with caffeine. This involves comparing the reaction times of PhD students without caffeine to first-year students with caffeine.

Features of Experiments

  • Participant assignment

  • Type of comparisons

  • Effect of individual differences

  • Risk of order effects

  • Statistical power

  • Number of participants needed

  • Confounding variables

  • Cases of best used when

Participant Assignment

  • Independent design: Different people in different groups. Different participants are assigned to different experimental groups.

  • Paired design: Same people in both conditions. The same participants are exposed to all experimental conditions.

Type of Comparisons

  • Independent: Between groups. Comparisons are made between different groups of participants.

  • Paired: Within the same group. Comparisons are made within the same group of participants under different conditions.

Effect of Individual Differences

  • Independent: High impact of individual variation, natural variations can provide added noise. Individual differences between participants can significantly impact the results, adding variability to the data.

  • Paired: Lower impact, individual serves as their own control. Using the same individuals as their own control reduces the impact of individual differences on the results.

Risk of Order Effects

  • Independent: No risk. There is no risk of order effects in independent designs since participants are only exposed to one condition.

  • Paired: Risk of order effects, e.g., caffeine still in blood. There is a risk of order effects in paired designs, such as the carryover effects of caffeine consumption.

Statistical Power

  • More data is good.

  • Independent: Lower statistical power. Independent designs generally have lower statistical power compared to paired designs.

  • Paired: Higher statistical power because differences between participants can be made larger when repeatedly measuring people. Paired designs tend to have higher statistical power because they reduce the variability due to individual differences.

Number of Participants Needed

  • More samples generally better.

  • Fewer participants needed for repeated measures design, but there is a minimum. Repeated measures designs typically require fewer participants compared to independent designs, but a minimum sample size is still necessary.

  • Avoid very small subgroups (e.g., two males in a class). Small subgroups can lead to unstable results and should be avoided.

Confounding Variables

  • Variable that has an unintended effect on the dependent variable. A confounding variable is an extraneous variable that influences both the independent and dependent variables.

  • Independent: Differences between groups in intelligence, age, or central nervous system activity can exacerbate the effects. These differences can confound the results and make it difficult to isolate the effect of the independent variable.

  • Paired: Order effects can play a role. Order effects, such as fatigue or practice effects, can confound the results in paired designs.

Cases of Best Used When

  • Independent: Large diverse sample, no risk of order effects. Independent designs are best used when there is a large, diverse sample and no concern for order effects.

  • Paired: Participant variability needs to be controlled, fewer participants are available. Paired designs are best used when controlling for participant variability is important and when there are fewer participants available.

T-Tests

  • Invented to brew beer better. T-tests were originally developed to improve the brewing process.

  • Used to compare two things to see if they are meaningfully different. T-tests are used to determine if there is a significant difference between the means of two groups.

  • Execution depends on whether it is an independent or paired design. The specific procedure for conducting a t-test depends on whether the design is independent or paired.

Independent T-Test

  • Comparing the mean reaction time between two independent groups. An independent t-test is used to compare the means of two independent groups.

Paired T-Test

  • Comparing the average reaction time within individuals across two conditions. A paired t-test is used to compare the means of two related groups (e.g., before and after measurements on the same individuals).

Choosing the Right Test

  • Depends on the hypothesis and practical factors like size, feasibility, and common sense. The choice of test depends on the research hypothesis, practical constraints, and logical considerations.

  • Independent t-tests: Comparing between two separate groups.

    • E.g., People who consume caffeine will have a faster reaction time than those who do not.

  • Paired t-tests: Comparing the same people before and after.

    • E.g., Individuals will have a faster reaction time after consuming caffeine compared to their own baseline.

  • Hypothesis is a guess, or an if and then statement. A hypothesis is a testable prediction or statement about the relationship between variables.

Research Question Example

  • Does caffeine improve reaction time more for first-year uni students or PhD students?

Steps

  • Identify the independent and dependent variables.

  • Determine the type of comparison.

Example 1: Independent Variable - Groups

  • Group 1: First-year uni students.

  • Group 2: PhD students.

  • Dependent variable: Reaction time.

  • Comparison between two groups.

  • Each participant belongs to only one group.

  • No within-individual comparison.

Example 2: Independent Variable - Condition

  • Condition 1: Reaction time of all students before having caffeine.

  • Condition 2: Reaction time of all students after having caffeine.

  • Reaction time is being compared within individuals, which results in a paired design.

Null Hypothesis

  • States that there is no effect or difference between the variables. The null hypothesis assumes that there is no significant relationship or difference between the variables being studied.

  • Default assumption that any observed differences are due to random chance. It serves as a starting point for statistical testing, assuming that any observed effects are due to random variation.

  • Expressed as H0H_0

  • We want to determine if the data observed comes from random chance or from the experiment and is represented by statistical tests attached to some form of distribution (e.g. t distribution, z distribution, or f distribution).

T Distribution

  • Histogram where the middle shows the most common result (zero). The t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small or when the population standard deviation is unknown. Its histogram is centered around zero, indicating that the most common result is no difference between the groups being compared.

  • Comparing the test statistic to the distribution. This involves calculating a t-statistic from the sample data and comparing it to the t-distribution to determine the probability of obtaining the observed results under the null hypothesis.

  • Can be based on p-values or critical values. The comparison can be based on p-values, which indicate the probability of observing the data if the null hypothesis is true, or critical values, which define the threshold for statistical significance.

Directionality

  • Some hypotheses have a direction, others are non-directional. Hypotheses can be directional (specifying the direction of the effect) or non-directional (simply stating that there is an effect).

  • Non-directional: Caffeine has any effect on reaction time.

  • Directional: Reaction time will be faster after having caffeine.

  • Non-directional tests split the p=0.05p = 0.05 value across the two ends of the distribution. Non-directional tests are two-tailed, meaning they consider both positive and negative effects, and the significance level is split between both tails of the distribution.

  • One-tailed tests refer to one side of the distribution. One-tailed tests are directional, considering only one direction of effect, and the entire significance level is concentrated in one tail of the distribution.

Paired Samples T-Test

Research Question

  • Does caffeine improve reaction time?

Null Hypothesis

  • Caffeine has no effect on reaction time; no significant difference in reaction times before and after having caffeine.

Alternative Hypothesis

  • Caffeine affects reaction time; there will be a significant difference in reaction times before and after.

Directional Hypothesis

  • Caffeine improves reaction time; times after consumption faster than before.

Design

  • Everyone experiences the same condition. Repeated measures. Reaction time is measured before and after.

Data

  • Participants A through R.

  • Reaction time before and after measured in milliseconds.

Steps (Manual)

  • Calculate the difference in reaction times under each condition for each participant.

  • Calculate the average and standard deviation of all those difference values.- Average=sum of valuesThe number of valuesAverage = \frac{sum \ of \ values}{The \ number \ of \ values}

  • Calculate degrees of freedom.- df=n−1df = n - 1

  • Calculate the test statistic.- t=Average of differencesStandard DeviationNt = \frac{Average \ of \ differences}{\frac{Standard \ Deviation}{\sqrt{N}}}

    • That simplifies to: t=Average∗nStandard Deviationt = Average * \frac{\sqrt{n}}{Standard \ Deviation}

Directional Hypothesis (One-tailed Test)

  • Caffeine will reduce reaction time (reaction times should be significantly faster).

  • Undertake a one-tailed test to see if there's any difference.

  • Work out the odds of getting that t-value with the relevant number of degrees of freedom.

  • Probability of randomly obtaining those results needs to be really small to suggest caffeine does actually alter the results.

Nondirectional Hypothesis (Two-tailed Test)

  • Two tailed, don't care what direction results are going in (faster or slower), just interested to know what change has been made. Can test using all collected data.

Interpreting the Results

  • P value around 1 in 12 million suggests that the tested T variable has a very slim chance of randomly influencing someone if caffeine is given to them. This is also known as an alpha point.

  • p < 0.05 so reject null hypothesis in favor of alternative hypothesis.- Supported. The data supports the hypothesis that it did cause reaction time to reduce.

  • However the previous results don't tell us what the direction is.- To know more need to use a One-Tailed Test hair test statistic, can calculate in a calculator around one in twenty five million. This again also enables you to reject a null hypothesis that caffeine does not improve reaction time.

    How to Write Up in APA.

  • Non-Directional Hypothesis Test- t(df) = value, p < 0.05

  • Directional Hypothesis- t(df) = value, p < 0.05

Do not be inconsistent with the number of decimal places in results. Recommendation. Stick to just two decimal places.

Independent Samples T-Test

Example

  • The example used here is about testing whether caffeine affects reaction time differently when examining a group of PHD students vs First Years.

Experimental Design

  • It involves measuring the reaction time of a group of PHD students after giving them nothing to ingest, and doing the same with a group of first years after giving them caffeine.

Outcome

  • There will be a significant difference between those from different experimental groups.

  • For an overall result it has to be hypothesized that giving a group with caffeine gives then faster reaction times.

Independent Sample

  • In this example all participants will only experience one condition. The variables under inspection are independent.

Setup Simulations:

Reaction time can be different between experimental groups. The reaction rate has two main components: Group 1 and Group 2. Group 1 average result= 254.00ms standard deviation: 21.8ms. Group 2 average results: 239ms; 33 standard deviation. Each group tested includes 9 people each.

Calculations:

  • Calculation the independent T-test statistic: Take group one mean, and minus group two mean. Divide by the square root of both the group 1 standard deviation squared / by its sample size and group twos equivalent result.- This can be reverse, the impact on the results will be that If it comes up negative, go for the abstract. If the P h result is higher: the value will need to show that group reaction time is higher vs first years. Swap around data results and analyse again. It's not actually going to affect test statistic.

  • Calculating the degrees of freedom: Degree of freedom group -1 + degree of freedom group 2 -1.- This means the total number for PHD + FIRST YEAR number of results -1 is 16. Since there's two sides to a hypothesis, so therefore two differing versions will test it.

Assumptions
* T- tests relies upon many assumptions. So if assumption criteria is not obeyed well then the test itself won't work. This does not apply to every exception. One is always the same, but we'll get back to it. The main important assumption being made, that the observations remain independent, not affecting each others results.
    *   Law Of Space.
    *   Each one should come from a totally seperate individual

There needs to be normality to the underlying distribution. What degree this is affected depends on test type being run. A test is said to be on normal dependent, but this really needs to be across the dependant variable within EACH group.

  • Homogeneity of variance this tends not to be a paired test/ test. What it