Z Scores

Technically, a Z score is the distance from the mean, measured in units of standard deviation.

It's very important to know how your specific teacher is teaching Z scores. In two sequential classes on statistics I was taught two completely different methods. In our first statistics class we were taught to look at Z scores on a Z score chart with had a Z score of 0 as equalling a .50 probability. The chart looked like this.

Z Score Chart - cumulative

However, in the second class, the Z chart instead uses Z to measure the distance from the mean (center point). So on their chart, if you have a Z score of 0, that results in a 0 value (instead of a .50 probability), because you are 0 distance from that mean centerpoint.

Z Score Chart - distance from center

In useful terms, we know we use a chart to determine the value of something based on a sample and the mean and standard deviation of that population. If they had to provide a chart for us for every single mean and standard deviation combination there would be millions of charts!

Rather than doing that, they provide in essence a "conversion factor". They make a chart for just one set of figures - that is, for a mean of 0 and a standard deviation of 1. Then they help us convert any other graph we look at to use this table, by using Z as our conversion tool.

That conversion factor looks like this:

Z = (X - mu) / sigma

So by figuring out that Z, we can then look at one standard chart and see what our probability answer is.

So let's say we have a standard normal probability distribution - a mean of 0 and a standard deviation of 1. In our example we'll say this is the temperatures in a cold storage freezer which needs to stay at about 0F. They had an average (mean) of 0F and distributed normally around it.

Standard deviation Chart

Using the second chart method, if we want to figure out the distance from the mean of a given sample of 1F is, we plug 1F into our formula, Z = (1/0)/1 = 1. We get a result, on the distance-from-mean chart, of .3413. That is the area of the curve between the 0 and the 1 mark on the horizontal axis.

OK now let's say we have a different freezer. This new freezer is supposed to stay at 20F. Again we plot out how well the freezer does in staying around 20F.

Standard deviation Chart

Now if we didn't have Z values we would need an entirely different chart because the numbers are different. However, due to the beauty of the Z value, we can use the exact same chart to figure this out even though our range of temperatures are centered around 20F. Let's say we want to look at 21F this time - so again 1 degree away from the mean. So we use that same formula - Z = (21 - 20) / 1. That is a Z value of 1 which on the same chart is .3413.

So the Z value lets us use the same chart to determine results, even though our source graphs have different means or standard deviations.

In terms of *probability* - the probability of something being at the mean is .50. so if you wanted to figure out what the probability is of something being anywhere from the leftmost side up through the 1F mark, you take this value you just found - .3413 - and you add it to the .50 which represents everything up to the mean. So the total would be .8413. which, looking at that other "cumulative style chart", is indeed the value you find for a Z of 1.

This following information is primarily for people who are trying to handle a professor change - where they're going from learning the "cumulative Z chart" way to now using the "distance from mean Z chart" way. In my statistics 1 book we used a Z table which provided cumulative values. That is VERY different from the statistics 2 book which provides distance-from-mean values. So I'm trying to make sure explain how these two charts are different and therefore how to use the second distance-from-mean chart.

So. Let's start with the basics. Let's say you have a normal chart. You want to know what proportion of entries are found up to the mean (center) point -

Standard deviation Chart

That is always .5. So that is important to know as a foundation step.

Now, let's say that we want to find out the probability of the chart having a value up to the 1 value. In the OLD book we would plug in that 1 value into the formula and get this result:

Standard deviation Chart

So the statistics 1 book would give us the total value for everything up to 1. The chart would give us the value of .8413.

So far so good?

Now, on to the statistics 2 book. When we look at the chart in the statistics 2 book, it does not tell us that total value number. Instead, it tells us the value for the **distance from the mean**. So what it is telling us is this -

Standard deviation Chart

So the new chart tells us the value of .3413 which is solely the information in the green area - from the mean to our target value.

How is that helpful, you might ask? Well we *already* know the value of all the area up to the mean. That was in the very first chart in this post area I'm making. That was the .5 value. So we add .5 (the value of everything up to the center point) plus .3413 (the value of the center point up to our target number). Those two added together equal .8413. So .8413 is what the old table would have told us.

They are giving us the same details, it's just this new chart makes us work a little more to get that information.

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