Introduction

In descriptive statistics, population mean, median, mode and range are simple statistical measures used to compare sets of data.

The mean is the arithmetic average of a set of numbers, and is probably the statistic most often used to describe a set of data. To calculate the mean, add up all the values in the set and then divide by the number of values in the set.

The median is the “middle” value in a set of values. To calculate the median, first arrange all the values in order from smallest to largest. If there is an odd number of values in the set, the median is the value that is in the middle after you have arranged them. If there is an even number of values, then there will not be one single middle value; instead, the median will be calculated as being halfway between those two middle values.

The mode is simply the value that occurs most often within a set ofvalues.

The Normal Distribution

In a normal distribution, the x value corresponding to z=.075 can be found by using a table of standardized normal deviates. The table value for z=.075 is 1.645. This means that the x value is 1.645 standard deviations above the mean.

The Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is equal to 0 and the standard deviation is equal to 1. Because it is such a important distribution, it often given it’s own name and symbol; the standard normal distribution is also called the z-distribution.

The z-distribution is continuous (meaning that it can take on any value), symmetric (meaning that it looks the same on either side of the mean), and bell-shaped (meaning that it has a single peak).

A random variable that follows a standard normal distribution is called a z-score. The z-score tells you how many standard deviations away from the mean a data point is.

The x-value corresponding to z=0.75

To find the x-value corresponding to z=0.75, we first need to determine which side of the mean the value falls on. Since z=0.75 is positive, we know that the x-value will be greater than the mean. We can then use a table of standard normal deviations to find the exact value.

The table tells us that z=0.75 falls between 0.67 and 0.80. We can interpolate to get a more precise value:

x = (0.67 + 0.80) / 2 = 0.735

This means that the x-value corresponding to z=0.75 is 0.735 standard deviations above the mean.

Conclusion

In order to find the x value corresponding to z=0.75, we need to first standardize the distribution. To do this, we take the z-score and multiply it by the standard deviation, and then add the mean.

z = (x-mean)/standard deviation

0.75 = (x-20)/10

7.5 = x – 20

x = 27.5