When we have large sets of data,

then we have another way of finding the position of a specific observation.

This is known as the Z-score.

By using the Z-score we can find the proportion of data points that

are less than a specific value.

The z-score is calculating using this formula.

Here x is the variable of interest.

Mu is the mean for the population and sigma is the standard deviation.

For example, if you were graduating from a college with a degree in business.

You get a job offer and would like to know how you offer stacks up against others.

All other peers from colleges get job offers similar to yours.

If all you knew is the median, then you can tell if you're in the top 50% or

the lower 50%, but are you at 51% or 95%?

We can calculate the position of your offer by calculating its Z-score.

Now, let's expand on this example we were using before.

You interview for a job and get an offer, and

would like to know how competitive is your offered salary.

All you have is some data based on published reports on websites.

Let's look at such websites, I'm using payscale.com.

Searching for a job title such as business analyst gives the following information.

The median is 54,030 and the standard deviation, about 8,900.

You get an offer of 60,000.

This is above the median, but how does the salary stack up compared to others?

To know where the salary falls, you can determine its percentile.

To do this, we need to first find out how many standard deviations

the offered salary is from the mean or the median?

This is known as the Z-score and it's calculated by taking x minus the mean and

dividing it by the standard deviation.

For you, the x is the salary offered to you of 65,000.

An average salary is 54,030.

Standard deviation, 8,600.

This means you received an offer, which is 1.27 standard deviations above the mean.

How can we tell where we stand in comparison to the rest of the people

getting such offer?

Is the salary in the top 70%?

Top 99%?

What exactly could it be?

Now, let me share with you some very useful properties of a bell curve.

For large datasets,

we often observe that many values cluster around the mean or the median.

So if then we create a histogram of the data,

we get a distribution that represents a bell shape, a symmetrical curve.

When this happens, according to the empirical rule,

68% of all observations will fall within one standard deviation of the mean.

95% will fall within two standard deviations and

99.7% of all observation will fall within three standard deviations.

Knowing this rule of one standard deviations, two standard deviation,

three standard deviation.

Known as the empirical rule will always be a handy way of figuring

out where an observation of interest falls in comparison to the mean or the median.

So given a specific observation, the offer we receive and

knowing something about this type of job starting salaries.

We were able to find our relative place in the group.

The Z- score, when positive tells you that the value is above the mean.

And when it's negative, it's below the mean.

Furthermore, it tells you how many standard deviations are you above or

below the mean.