Associating p-value to z-score on population distribution

"p-values" are a core concept of statistics. In simple terms, they representing the "probability" of a given event occurring. They enable us to determine the "statistical significance" of the given event, which will be used in conjunction with confidence intervals and hypothesis tests.

p-value stands for probability-value, and is a number between 0.000 and 1.000 representing the probability of an event occurring in a population.

When able to do so, the best way to find a p-value is with a z-score (also referred to as "z-test statistic").

To find our p-value for the below scenario prompted in z-scores and normal distributions...

What percentage of the population has an IQ score less than 105?

...we're first going to calculate the z-score for the IQ score of 105, and then we'll find the corresponding p-value in the z-table!

Calculating the z-score

Therefore, its formula looks like this:

What do each of these variables mean?

x is the data point that you want to obtain the z-score of.
μ is the mean value of the given population.
σ is the standard deviation of the given population’s values.

Plugging in the mean (μ) and standard deviation (σ)

In z-scores and normal distributions, we stated the following about the distribution of IQ scores:

"Well, it is known that all IQ scores follow on a normal distribution like this...

...with a mean of 100...

...and a standard deviation of 15."

Therefore, we can plug in 100 for μ...

...and 15 for σ!

Plugging in the data point (x)

Above we stated that we got an IQ score of 105.

"You've always been curious how smart you are compared to the average person, so you decide to take an IQ test. After taking the test, you obtain a score of 105."

Therefore, we'll plug in 105 for x.

Solving for z-score

When we solve this out, we get 0.33!

How to associate a p-value to your z-score

Cool! You've got a z-score of 0.33! But in reference to our prompt...

What percentage of the population has an IQ score less than 105? do we utilize that to claim that we're "smarter than __% of the population"? We still need to find that "__%"!

We'll use z-table! (Don't sleep on this table. It's super clutch.)

All we need to do is find "0.3" in the left-hand column of the right table (representing 0.33)...

...and then "0.03" in the top row (representing 0.33)... locate our p-value of 0.6293!

Understanding your p-value visually

Notice how at the top of the z-table, the area to the left of the "z" on the x-axis is filled in?

That's because...

The p-values in the z-table are telling you what the area to the left of your z-score is!

So in the case of our 105 IQ score on the distribution of all IQ scores...

...this means that 0.6293, or 62.93% of the area under the curve occurs to the left of our 105 IQ score!

In other words, to answer our original question...

What percentage of the population has an IQ score less than 105?

...we're "smarter than 62.93% of the population"!

What if we're dealing with a sample, not a data point?

Up until now, we've just been considering one data point (our IQ score of 105).

What if instead, I asked you the following:

What's the probability of finding a sample of 30 people will a mean IQ score of less than 105?

Same IQ score of 105, but now we're dealing with trying to find a sample of people who collectively have a mean IQ score of 105. We'll still use z-scores here... but how does this change up things from before when we were dealing with only one person?

It's actually not as different as you'd expect... click here to learn more!

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