Probabilities are calculated using technology. There are instructions given as necessary for Google Sheets, many of which will work fine in other modern spreadsheet applications as well.
The NORM.S.DIST() function assumes $\mu =0$ and $\sigma=1$ and will give you the probability/area to the left of a z-score that you input into the function. For example, if you type =NORM.S.DIST(2.4), the cell will show the value 0.9918024641 (we generally only need 4 decimal places). This tells us that the probability to the left of $z=2.4$ is 0.9918.
In most situations, the mean will not be 0, and/or the standard deviation will not be 1. In these cases, you can convert the value for which you want the probability (what we sometimes will call the “$x$-value”) into a $z$-score with the standard formula below and plug that $z$-score into the NORM.S.DIST() function.
$$z = \frac{x-\mu}{\sigma}$$
NORM.S.DIST
Returns the value of the standard normal cumulative distribution function for a specified value.
Sample Usage
NORM.S.DIST(2.4)
NORM.S.DIST(A2)
Syntax
NORM.S.DIST(x)
x
– The input to the standard normal cumulative distribution function.
Notes
- The “standard” normal distribution function is the normal distribution function with mean of
0
and variance (and therefore standard deviation) of 1
.
Converting between $z$-scores and $x$-values
We have seen that to convert an $x$-value whose distribution is normal with a mean $\mu$ and a standard deviation $\sigma$, we can use the standard $z$-score formula.
$$z=\frac{x-\mu}{\sigma}$$
We often times need to convert a $z$-score back into an $x$-value. A bit of algebraic manipulation on the $z$-score formula gives us the following formula to convert a $z$-score into an $x$-value. We show this in part 3 of Example 6.8 below.
$$ x = z \cdot \sigma + \mu$$
When we need to take a probability and find the associated $z$-score, we can use the NORM.S.INV() function. Note, this is the reverse of what we did previously with the NORM.S.DIST() function, where we had a $z$-score and we needed to find the probability.
In part 3 of Example 6.8, we used =NORM.S.INV(0.9) to find that 1.28 was the $z$-score that had 0.9, or 90% to the left.
1.28 is then said to be the 90th percentile.
NORM.S.INV
Returns the value of the inverse standard normal distribution function for a specified value.
Sample Usage
NORM.S.INV(.75)
NORM.S.INV(A2)
Syntax
NORM.S.INV(x)
x
– The input to the inverse standard normal distribution function.
Notes
- The “standard” normal distribution function is the normal distribution function with mean of
0
and variance (and therefore standard deviation) of 1
. x
must be greater than 0
and less than 1
or a #NUM!
error will occur.