# Z Score Table

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### Z-Score Table:

In Statistics, we come across the word ‘random variables’ very often. The term random variables is not like the variables we encounter in algebraic expressions. From a random experiment, the set of possible values obtained are known as the random variables. Random variables are the set of all the possible values and out of the set it can take any value randomly. For instance in an experiment of tossing a coin, there are 2 possibilities: getting ‘heads’ or ‘getting tails’. Let us assume the value of ‘0’ for heads and ‘1’ for tails and have a random variable ‘X’. Then we see that the two random events are heads and tails. It is written as X = {0, 1}. Sample Space is the set of all the values of the random variable. These random variables can be discrete or continuous. The random variable in the normal equation is the normal random variable.
The normal distribution function is given by the equation as shown below: Z-Score Definition:
A z-score value is measurement in Statistics which indicates a given score’s relationship to the mean.  Out of several scores, the z-score indicates how many standard deviations a given particular score is from the mean. The z-score value is also known as the Standard score, z-values, and normal scores. Since the normal distribution is also known as the ‘z distribution’, hence we use the letter ‘z’ for the score.
Interpretation of the z-score:
If the z-score is 0, then it implies that the score is the same as the mean.
If the z-score is negative, then it implies that the score is lesser than the mean.
If the z-score is positive, then it implies that the score is greater than the mean.
If the z-score is 1, then it implies that the score is 1 standard deviation greater than the mean.
If the z-score is -1, then it implies that the score is 1 standard deviation lesser than the mean.
If the z-score is 2, then it implies that the score is 2 standard deviation greater than the mean.
If the z-score is -2, then it implies that the score is 2 standard deviations lesser than the mean.

Z-score formula:
Given the mean and the standard deviation, we can calculate the z-score value from the formula shown below: Example: In an exam, a student receives a score of 88 when the mean of the class is 76 with a standard deviation of 10. How much is the z-score?

Given the student’s score, x = 88
Mean, μ = 76
Standard deviation, σ = 10
From the z-score formula, we get the equation: z = (x – μ)/ σ ==> z = (88 – 76)/ 10
z = 12/10 = 1.2
Therefore the z-score in this case = 1.2
With this z-score we can say that the student has scored above average (mean), with a distance of 1.2 from the average score.

Mean:
Mean is the average of the given scores. From a set of values, mean can be calculated by adding all the values and then dividing by the number of values in the set as shown in the figure below: Example: What is the mean of the values: 4, 7, 11, 10, 8?
Given 5 values.
Mean = Sum of the values/ Number of values
So mean = (4+ 7+ 11+ 10+ 8)/ 5 = 40/5 = 8
Hence the mean of the given values is 8.

### Standard Deviation:

The measure of finding how spread out or distributed the values are in a given set of values is the Standard Deviation. It is expressed using the Greek symbol σ (sigma).
Standard Deviation can be easily calculated by finding the square root of the Variance.

### Normal Probability Distribution:

Normal Probability Distribution, also known as the ‘Bell Curve’ is a graph which depicts a standard normal distribution. The name Bell Curve is given because of its resemblance to a bell shape. Bell curves are extremely important and are very commonly used throughout the studies in Statistics. The two numbers that are necessary for plotting this Bell curve are the Mean and the Standard Deviation.

The Bell curve (figure shown below) for a certain set of data always has the mean located at the center. This is the also the point where the curve reaches the highest point, known as the ‘top of the bell’. The Standard Deviation determines how much the given data set values are spread out. If the Standard deviation is larger, then the bell curve will be more spread out. A Bell curve is symmetric and hence when folded at the mean, we get 2 equal halves. As shown in the figures below, it is estimated that approximately 68% of all the given data lies within 1 standard deviation of the mean. And, about 95% is covered within 2 standard deviations of the mean, and about 99.7% is covered within 3 standard deviations of the mean.   Negative Z-Score Table:
The Negative Z-Score Table consists of all the negative z-scores. These negative z-scores are all to the left of the mean. Left of the mean is also considered as being below the mean.  Positive Z-Score Table:

The Positive Z-Score Table consists of all the positive z-scores which are to the right of the mean. These positive z-scores are to the right of the mean. They are also considered to be above the mean.  Example:
In the Science exam, the mean of the scores of the class is 80 with a standard deviation of 12. Jack scored 98 on the exam. What is the approximate percentage of students who scored lesser than Jack’s score? Given the student Jack’s score, x = 98
Mean, μ = 80
Standard deviation, σ = 12
From the z-score formula, we get the equation: z = (x – μ)/ σ ==> z = (98 – 80)/ 12
z = 18/12 = 1.5
Therefore the z-score in this case = 1.5

Since the z-score is a positive number, we look at the Positive Z-Score table for the percent of students who scored within 1.5.
As highlighted in the figure on the right, the value is 0.9332, which implies 93.32% of students scored lesser than Jack.