Chi Squared Test
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Chi Squared Test
Why Chi Squared?To test to see if, when we collect data, is the variation we see due to chance or due to something else?
This is the Chi Squared Test FormulaGreek letter X is ChiDont get confused- X2 is the name of the whole variable- you dont have to ever take the square root of it or solve for X.
Coin Flipping ExampleIf you flip a coin 100 times and get this data:62 Heads38 TailsThe question we wonder about this data:Is this due to chance? Or. . .Is this not due to chance?For example:Perhaps there something wrong with the coin?Perhaps there is something wrong with the way I am flipping the coin, etc.?Chi Squared test lets us answer this!!
Null HypothesisThe Chi Squared Test begins with the Null HypothesisThe Null Hypothesis says: There is no significant statistical difference between the observed and expected frequencies (i.e. the differences we see are simply due to chance).For a coin flipping experiment:The expected values are 50 heads, 50 tails.The observed values are 62 heads, 38 tails.Question we are trying to answer: are the observed values due to chance or. . . not due to chance?
Degrees of Freedom and Critical ValuesWe need to define and understand these terms before we can use the Chi Squared test.The whole point of a Chi Squared test is to either reject or fail to reject (accept) the Null Hypothesis. Key is to exceed or not exceed your critical value. But first we have to figure out which number it is in this chart.
Degrees of FreedomBecause we are comparing outcomes, we need at least two outcomes in our experiment. We are flipping coins so we have two outcomes- heads or tails. To get degrees of freedom we simply subtract one from the two possible outcomes. 2-1= 1Therefore, in this experiment we have 1 degree of freedom
Critical ValuesNext thing you are looking for is a critical value. We will always use p = 0.05 value.This means we are 95% sure we are either failing to reject the Null Hypothesis or rejecting our Null Hypothesis.Critical Values can vary. If we want a higher degree of certainty that our results are true, we can use p= 0.01 value and then we would have 99% certainty, but 95% certainty is used by most scientists.For 1 degree of freedom at p=.05 the critical value = 3.84.
Null HypothesisNull Hypothesis says:There is no significant statistical difference between the observed and expected frequencies (i.e. the differences we see are simply due to chance).We either reject or fail to reject (accept) the null hypothesis.Reject the Null Hypothesis: This means there is a statistical difference between the observed and expected frequenciesFail to Reject (accept) the Null Hypothesis: This means there is no statistical difference between the observed and expected frequenciesIn this case the critical value is 3.841If your Chi Squared is greater than 3.841 you reject the Null HypothesisTherefore, there is something aside from chance that is causing us to get more heads than tails. If your Chi Squared is less than 3.841 you fail to reject (accept) the Null Hypothesis.This is usually what happens unless you have something that is impacting your results
Practice- Coin Toss50 tossesExpected: Heads: 25 Tails: 25Observed: Heads: 28 Tails: 22Do it for headsDo it for tailsWhat is our Critical Value?2-1 = 13.84If our Chi Squared is greater than the critical value we reject our Null HypothesisIf our Chi Squared is less than the critical value we fail to reject (accept) our Null Hypothesis This means there is no statistical significance between what we observed & what we expected. What we got is due to chance- nothings weird about the coins or the way we tossed them.
Practice- Dice36 diceExpected: 6 of eachObserved:
Practice- DiceChi SquaredDegrees of Freedom6-1=5Critical Value: 11.07Fail to reject (accept) the Null HypothesisNo difference between obs and expWhat we see is due to chance!