Z-Scores, Statistics Problem Example

Where x is a measured value, ? is the mean ? is the standard deviation of all measured values.

when x= 21 you get a positive answer , it is positive because 21 is greater than 17.

z = (21-17)/3 = 1.33  This means 21 is 1.33 standard deviations from the mean in the positive direction.

Another day when you change the formula and have  x = 12

z = (12-17)/3 = -5/3 = -1.67  you will get a negative answer. 12 is 1.67 standard deviations below the mean which moves in a negative direction.

The third day when x = 17, The results expected when x in this case is the expected value or the mean therefore zero standard deviations from itself. z= 0

Four uses of z-scores

1. Interpret raw scores, when you covert z-score from a raw score, the z-score indicates the number of standard deviations above or below the mean the raw score is. When you have a positive z-scores, like on Eric’s first day, it indicate that the raw score was above the mean;. When it is negative, like on the second day, the z-scores indicate that the raw score was below the mean.
2. Compare different variables when raw score is converted, it can standardize different scores, such as all three days where each measured different standard deviations. It gives comparable data when knowing the time it takes to get from work to home.
3. Compute relative frequency by finding out the z-score. Giving information about the standing of a z-score in relation to the distance from the mean, different times it takes to get home, and the standard deviations.  The relative frequency of the z-score is the same on al l normal distributions.  It can be computing using the proportion of the area under the distribution curve.
4. Describe sample means: In getting a sample from a population, in order to test out situations. Sample means go along with population means, the distribution is normal if the population is normal, and should be close to the expected value of x. Using the formula as with all the Eric days , z-scores calculated describe the position of any sample in the distribution.

The normal curve and z-score have a special relation , a common statistical way of standardizing data on one scale so a comparison can take place is using a z-score. Each z-score corresponds to a point in a normal distribution, although not an exact representation it is close,  and as such is sometimes called a normal curve since a z-score will describe how much it deviates from a mean or specification point measured in standard deviations. It becomes easier to compare different parameters because of the standardization the z-score provides.