Z Score, Z Statistics and P value

What is a Z score or Z statistics?

Firstly Z score and Z statistics is the same. It is a measure of the number of standard deviations a data point is away from the sample mean. Mathematically this is written as:

`z = (x - bar x) / sigma`

where
`z` = Z score or Z statistics
`x` = data point
`bar x` = sample mean
`sigma` = sample standard deviation

The mathematical equation of standard deviation is:

`sigma = sqrt((x - bar x)/N)`

where

`N` = number of sample points

What is P value?

The P value is the probability that a particular value assuming the null hypothesis is true.

Conventionally a P value of less than 0.05 is grounds for rejecting the null hypothesis.

For the lower-tailed test or left-tailed test, the P value is expressed as:

`P = Pr(TS <=ts | H_0) = cdf(ts)`

And for the upper-tailed test or right-tailed test, the P value is expressed as:

`P=Pr(TS>=ts |H_0) = 1-cdf(ts)`

And the P value for a two-tailed test is

`P = 2 xx min{Pr(TS<=ts |H_0), Pr(TS>=ts |H_0}`

where
`P` = P value of an observation
`TS` = test statistics
`ts` = observed value of test statistics from your sample
`cdf()` = Cumulative distribution function for the test statistic
`Pr(condition|H_0`) = probability of the observation if the null hypothesis `H_0` is true.
`min` function to select the minimum value

How to get P value from Z score?

Calculating Z score depends on the problem analysis.

Fortunately calculating P value from Z score is easy in Excel. Assuming the Z score has been calculated z then

p_value = NORM.S.DIST(z, TRUE)

What about T score / T statistics?

Likewise calculating T score or T statistics depends on the problem analysis. But it is related the t distribution.

Then in Excel the

p_value = T.DIST(t_score, degrees_of_freedom, TRUE)

I thought I write about this before launching into writing hypothesis testing formulae with LAMBDA where you will see z scores and t scores popping up. In the meantime, you might be interested in brushing up on your statistics in your DC-DEN!