Goodness of Fit (Edexcel A Level Further Maths: Further Statistics 1): Revision Note

Goodness of fit

What is the difference between observed values and expected values?

• Goodness of fit is a measure of how well real-life observed data fits a theoretical model

  • For example, modelling a coin as fair then flipping it 20 times  

    • You may observe 13 heads

    • You would expect 10 heads

• Observed ($O_i$) and expected ($E_i$) values can be shown in a table

  • For example, rolling a fair die 60 times ($N=60$)

    Outcome	1	2	3	4	5	6
    $O_i$	12	7	8	10	14	9
    $E_i$	10	10	10	10	10	10

    • Note that 

• How different do observed and expected values need to be before the model is not a good fit? 

  • You can do a hypothesis test to reach a conclusion

What are the null and alternative hypotheses?

• ${\mathrm{H}}_0:$ There is no difference between the observed and the expected distribution

• ${\mathrm{H}}_1:$ The observed distribution cannot be modelled by the expected distribution

• Let $\alpha \%$ be the significance level

How do I calculate the goodness of fit?

• First, combine any columns for which expected values are less than 5 until they are greater than 5

  • For example

    Score	1	2	3	4
    $O_i$	15	6	4	1
    $E_i$	12	8	4	2

    • The expected value of 2 is less than 5 so combine the last two columns

    Score	1	2	3+
    $O_i$	15	6	5
    $E_i$	12	8	6

• Then calculate the goodness of fit, $X^2$, from the formula

  • 

• An alternative version of the formula that can be easier to calculate is

  • 

    • Where $N$ is the sum of all observed values

    • 

    • This is also the same as the sum of all expected values

• The larger $X^2$ is, the more different the observed values are from the expected values

What are degrees of freedom?

• The number of degrees of freedom, $\nu$, is equal to

  • The number of columns (after combining to get $E_i>5$) subtract 1

• If you also use the observed data to estimate a parameter, then you subtract 2 instead

  • For example, trying to estimate $p$ when comparing to a $\mathrm{B}\left(n,p\right)$ distribution

• You are subtracting the number of constraints (or restrictions)

  • This is the number of times you use the observed data to help form the expected data

    • This is always 1 from ensuring their totals match,

    • Then another 1 for each parameter estimated

How do I use the chi-squared distribution?

• Once you have calculated the goodness of fit, $X^2$

  • Compare it to the critical value ${\chi }_{\nu }^2$ from the chi-squared distribution

    • $\nu$ is the number of degrees of freedom

    • Tables of critical values are provided in the exam

    • You need the significance level, $\alpha \%$

    • All chi-squared tests are one-tailed

  • If $X^2<{\chi }_{\nu }^2\left(\alpha \%\right)$ then there is insufficient evidence to reject ${\mathrm{H}}_0$

    • This means there is no difference between the observed and expected distributions

    • In other words, "the expected distribution is a suitable model for the data"

  • If $X^2>{\chi }_{\nu }^2\left(\alpha \%\right)$ then there is sufficient evidence to reject ${\mathrm{H}}_0$

    • The expected distribution is not a suitable model for the data

• Alternatively, you can use your calculator to find the ${\chi }_{\nu }^2$ p-value

  • This is the probability of obtaining a chi-squared value of $X^2$ or more

  • If $p<\alpha$ then the result is critical (reject ${\mathrm{H}}_0$)