**What are the values of the mean and standard deviation of a standard normal distribution? (For each answer, enter an exact number.)**

**u=**

*σ* =

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**Explanation:**

**Mean (u)**=0 and **Variance ( σ )**=1 of a standard normal distribution

**So standard deviation = √1=1**

**What is Normal Distribution?**

Normal Distribution is a bell-shaped curve generated by many different probability distributions that approximate how data is normally distributed. It consists of three parameters: the mean, the standard deviation, and the cumulative fraction of data located at each value in the distribution.

The center of the distribution has a value with an associated probability density function, which can be visualized via a histogram. The shape and location of this curve is used to test the hypothesis that given random samples, their observations generally follow a Normal Distribution.

The probability distribution of a normal distribution is centered on the mean. The graph above represents a typical feature in normal distribution curves called the Bell curve. The data in the above graph is mostly concentrated around the mean. The more you move away from the mean the more the data distribution decreases.

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**What is Standard Deviation?**

Standard Deviation is a measure of the dispersion of a set of data values, a commonly used measure of statistical dispersion. It measures how spread out the data is. Standard deviation is always *between 0 and the maximum value*. The larger the standard deviation, the greater variation or dispersion in the data values.

Standard deviation is calculated by deducting each data point from the mean value, then obtaining the squared mean of the values that were different. This is known as the variance calculation. The standard deviation is found by taking the variance’s square root.

**Here is an example:**

In the first graph, there is less deviation as compared to the second graph. Assuming the first graph represents the wealth distribution of people living in third world countries and the second represents a superpower country. The wealth distribution of people living in third world countries does not vary much hence the reason for the less deviation from the mean.

The wealth distribution in the superpower countries has a greater deviation due to unequal wealth distribution.

**Here is another example:**

Find the standard deviation of the height of four trees that are **60 cm, 80 cm, 120 cm, and 180 cm** respectively.

The first step is to calculate the mean, or the average of all these values, by adding up each value individually and dividing the total by the total number of data points.

60 cm+80 cm+120 cm+180 cm=340 cm

340/4=85

**Mean**=85

The next step is to subtract all the data points from the mean

60-85= -25

80-85= -5

120-85= 35

180-85= 95

Positive numbers indicate that the data point is above the mean, while negative values indicate that it is below the mean. A value of 0 indicates that the data point and the mean are identical.

Next, you square each value and find its average to calculate variance.

(-25)^{2 }+ (-5)^{2} + (35)^{2} + (95)^{2 }=

625+ 25+ 1225+ 9025= 10900

10900/4=2725

**Variance**=2725

**Square root of Variance** = **√**2725

= 52.20

**Standard deviation**= 52. 20

**What is Standard Normal Distribution?**

The Standard Normal Distribution is a continuous probability distribution with a mean of 0 and a standard deviation of 1. It’s often used in hypothesis testing since the probability of getting any value from the distribution is 0.22849 or 4%. This means that if you test several hypotheses using the Standard Normal Distribution, and 95% of them are true, then 95% of your tests will show the predicted result.

The vertical axis represents the values between negative infinity and positive infinity, and the horizontal axis represents real numbers between -1.and +1.0. The area under the curve is equal to 1.0, so the Standard Normal Curve is a normal probability distribution curve.

From the above graph, you can determine the characteristics of standard deviation

*The first standard deviation is occupied by 68% of the values.**95% of samples are inside the second standard deviation.**The third standard deviation is within which 99.7% fall.*

**What is Z-Score?**

The Z-Score is the result of a normal distribution multiplied by a number. It’s how many standard deviations an individual observation is from the mean. In science, this is also referred to as standard deviation (SD) or an SD score.

The formula for Z-score is:

**Z-point = Data point-mean/Standard deviation**

**Example**

The summary for cat results in the statistics exams are as follows: 10, 13,18, 25, 14, 18, 12, 9, 20. Calculate the mean, standard deviation, and z-score.

**Mean** = 10+ 13+ 18+ 25+ 14+ 18+ 12+ 9+ 20 / 9 = 15.4

Next, subtract the mean from the cat marks to get variance and standard deviation

10 – 15.4 = -5.4

13 – 15.4 = -2.4

18 – 15.4 = 2.6

25 – 15.4 = 9.6

14 – 15.4 = – 1.4

18- 15.4 =2.6

12 – 15.4 = -3.4

9 – 15.4 = -6.4

20 – 15.4 = 4.6

**Variance** = (-5.4)^{2 }+ (-2.4)^{2 }+ (2.6)2 + (9.6)2 + (-1.4)^{2 }+ (2.6)^{2 }+ (-3.4)2 + (-6.4)^{2 }+ (4.6)^{2 }

= 29.16 + 5.76 + 6.76 + 92.16 +1.96 + 6.76 +11.56 + 40.96 + 21.16 / 9 =24.03

**Standard deviation **= square root of variance

= **√**24.03

** = 4.90**

The Z-Score reveals the data point’s position in relation to other data points. The z-score will indicate, in units of your standard deviation, how distant a point is from the mean. Determine the z-score for each point now:

**FOR X = 10**, Z= -1.10

**FOR X =13,** Z= -0.48

**FOR X =18,** Z= 0.53

**FOR X = 25,** Z = 1.95

**FOR X = 14,** Z = -0.28

**FOR X = 12**, Z = -0.69

**FOR X = 9**, Z = -1.30

**FOR X =20**, Z = 0.93

Positive numbers signify that the point is above the mean, while negative values indicate that it is below the mean. The difference between the mean and each datapoint can be calculated by multiplying each value by the standard deviation.

In this article on we have learnt how to look and recodnize normal distribution. We have also looked at standard deviation and explored the z-score all with a solved examples. If you’d like to learn more about standard deviation, standard normal distribution and other statistical concepts you explore a career in statistics on data analysis.

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