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The normal distribution function is given by the equation as shown below:

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.

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.

Given the mean and the standard deviation, we can calculate the z-score value from the formula shown below:

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 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:

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 can be easily calculated by finding the square root of the Variance.

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.

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:

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