[Math] How to convert non-normal distribution to a normal distribution

statistical-inferencestatistics

I have the below graph I drew using histogram in Excel but when I got my data and graphed it, certainly it is not a normal distribution. My assignment required me to gather some data and draw normal distributions and confidence intervals etc but I don't know what to do because my graph is not a normal distribution?

Should I fabricate some data to create a normal distribution or is there a legit way to create the normal distribution from this data? I read from a Google search there is Box-Cox transformation to create a normal distribution from non-normal data but being in a beginner stats class as an engineer I don't have that level of knowledge to achieve that.

Please help. Here is what my graphs look like:

enter image description here

If I should create data to resemble a normal distribution then how do I generate random data for each histogram range with a frequency to generate a sample set with normal distribution?

Best Answer

Obviously the data does not appear normally distributed. If you are restricted to using the existing data then you might take small sub-samples and calculate the averages. The distribution of this average should appear more like a normal distribution -- as per the central limit theorem.

If you want to generate some new normally distributed data, you can use the random number generator in Excel. Generate some uniformly distributed random numbers using the RAND() function, then apply the inverse normal distribution to each one -- NORMSINV(). Then you will have a sample from a normal distribution with mean $0$ and variance $1$. If you want a mean $\mu$ and variance $\sigma^2$, then transform as $x \mapsto\mu+ \sigma x.$

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[Math] How to find the probability range/scale for a histogram

Assuming that the different scales are multiples of each other and that the original plot has a vertical axis in the count scale, just use the definitions of the probability and probability density scales.

Let the sample size be $n$ and the values for each bin are $c_i$, $p_i$, and $d_i$ (for $i=1,2,\ldots,n$) for the count, probability, and probability density scale. That each are multiples of the other we can write $p_i=w c_i$ and $d_i=v c_i$.

We know that the sum of the probabilities and the sum of the areas are 1:

$$\sum_{i=1}^n p_i=\sum_{i=1}^n w c_i=w \sum_{i=1}^n c_i=w n=1$$ $$\sum_{i=1}^n 35 d_i=\sum_{i=1}^n 35 v c_i=35v \sum_{i=1}^n c_i=35 v n=1$$

So $w=1/n$ and $v=1/(35n)$.

The maximum count occurs at 28 so "nice" tick marks can occur at 0, 5, 10, 15, 20, 25, and 30. The maximum probability occurs at $28 w=28/100=0.28$ so "nice" tick marks are 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, and 0.30. The maximum probability density occurs at $28 v=28/(35*100)=0.008$ so "nice" tick marks will be 0.000, 0.001, ..., 0.009.

Now we find the corresponding counts to those sets of tick marks. That will be 0, 5, 10, 15, 20, 25, 30 for the probability axis and 0, 3.5, 7, 10.5, 14, 17.5, 21, 24.5, 28, 31.5 for the probability density axis.

Using commands from R one has

par(mai=c(1,1,1,1.2), xpd=TRUE)
plot(c(0,775), c(0,32), type="n", xlab="", ylab="Count", 
  axes=FALSE, las=1)
axis(1, c(0:4)*100, pos=0)
text(200, -5,"Reaction time (ms)")
axis(2, las=1, pos=0)
axis(4, c(0:6)*5, c("0.00", "0.05", "0.10", "0.15", "0.20", "0.25", "0.30"),
  las=1, pos=450)
text(550, 15, "Probability", srt=90)
axis(4, c(0, 3.5, 7, 10.5, 14, 17.5, 21, 24.5, 28, 31.5), 
  c("0.000", 0.001, 0.002, 0.003,0.004, 0.005, 0.006, 0.007, 0.008, 0.009),
  las=1, pos=650)
text(775, 15, "Probability density", srt=90)

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[Math] How to find the probability range/scale for a histogram

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