use the concepts of sampling error and zscores to explain th

use the concepts of sampling error and zscores to explain the concept of distribution of sample means

Solution

What is a Z-Score?

A z-score is a measure of how many standard deviations below or above the population mean a raw score is. A z-score is also known as a standard score and it can be placed on a normal distribution curve. Z-scores range from -3 std devs (which would fall to the far left of the normal distribution curve) to +3 std devs (which would fall to the far right of the normal distribution curve). In order to use a z-score, you need to know the mean and the population standard deviation . Simply put, a z-score is the number of standard deviations from the mean a data point is.

Z-scores are a way to compare results from a test to a “normal” population. Results from tests or surveys have thousands of possible results and units. Often, those results can seem meaningless. For example, knowing that someone’s weight is 150 pounds might be good information, but if you want to compare it to the “average” person’s weight, looking at a vast table of data can be overwhelming (especially if some weights are recorded in kilograms). A z-score can tell you where that person’s weight is compared to the average population’s mean weight.

Z Score Formula: One Sample

The basic z score formula for a sample is:
z = x – /
For example, let’s say you have a test score of 190. The test has a mean () of 150 and a standard deviation () of 25. Assuming a normal distribution, your z score would be:
z = x – /
= 190 – 150 / 25 = 1.6.
The z score tells you how many standard deviations from the mean your score is. In this example, your score is 1.6 standard deviations above the mean.

Z Score Formula: Standard Error of the Mean

When you have multiple samples and want to describe the standard deviation of those sample means (the standard error), you would use this z score formula:
z = x – / ( / n)
This z-score will tell you how many standard errors there are between the sample mean and the population mean.

Z scores and Standard Deviations

Technically, a z-score is the number of standard deviations from the mean value of the reference population (a population whose known values have been recorded).

A z-score tells you where the score lies on a normal distribution curve. A z-score of zero tells you the values is exactly average (red circle on the chart) while a score of +3 tells you that the value is much higher than average (blue circle on the chart).