# Chapter 4 : Variance and Standard Deviation

### Topics covered in this snack-sized chapter:

#### Variability arrow_upward

• The goal for variability is to obtain a measure of how the scores spread out in a distribution.
• A measure of variability accompanies a measure of central tendency as basic descriptive statistics for a set of scores.
• Variability can be measured with:
• Range.
• Interquartile range.
• Standard deviation.
• Variance.

#### Standard Deviation arrow_upward

• A measure of variability that describes an average distance of every score from the mean is known as standard deviation.
• The square root of the variance is also standard deviation.
• It shows variation about the Mean.
• Population Standard Deviation :
• If x1 , x2 , …….. xN denotes all N values from a population, then the population standard deviation is given by: Where, is the mean of population.

• Sample Standard Deviation (S):
• If x1 , x2 , …….. xN denotes all N values from a population, then the sample standard deviation is given by: Where, is the mean of sample.

#### Steps to Calculate Standard Deviation of a Sample arrow_upward

• Calculate the mean of the sample.
• Find the difference between each entry (x) and the mean. Square the deviations from the mean.
• Sum the squares of the deviations from the mean.
• Divide the sum by (n – 1) to get the variance.
• Take the square root of the variance to get the standard deviation.
• ##### Example:
• Find the Standard deviation of the data set given below:
• 1, 2, 3, 4, 5
##### Step 1:

 X   1 3 -2 4 2 3 -1 1 3 3 0 0 4 3 1 1 5 3 2 4

##### Step 2:
• Find the sum of • 4 + 1 + 0 + 1 + 4 = 10
• ##### Step 3:
• Find n – 1.
• n = 5
• n – 1 = 5 – 1 = 4
• ##### Step 4:
• Now we get the Standard Deviation using the formula:
• • #### Variance arrow_upward

• The square of the standard deviation, it is a description of how much each score varies from the mean.
• ##### Example:
• Determine the standard deviation and variance from following data:
•     30, 26 and 22

• Here, n = 3.
• ##### Solution:   30 4 16 26 0 0 22 -4 16 Mean = 26 0 32

• The Variance:
• = 32 ÷ 2 = 16

• The Standard Deviation:
• #### Understanding Standard Deviations arrow_upward

• Greater means more dispersion of data.
• Below are the two data sets with same mean but different standard deviations.
•  #### Coefficient of Variation arrow_upward

• It is the measure of relative variation.
• It shows variation relative to the mean.
• It is used to compare two or more sets of data measured in different units.
• • Where,
• = Sample Standard Deviation. = Sample Mean.

#### Z – Score arrow_upward

• Z – Score is the difference between the value and the mean, divided by the standard deviation.
• The value of the z-score tells exactly where a raw score is located relative to all the other scores in the distribution.
• A score that is located two standard deviations above the mean will have a z-score of +2.00.
• The z-score is often called the Standardized Value.
• Useful in identifying outliers (extreme values).
• Outliers are values in a data set that are located far from the mean.
• The larger the Z – Score, the larger the distance from the mean.
• A Z – Score is considered an outlier if it is • Or

• #### Negative Z – Score arrow_upward

• Z- Score is negative if a data value is less than the sample mean.

• #### Positive Z – Score arrow_upward

• Z- Score is positive if a data value is greater than the sample mean.

• #### Zero Z – Score arrow_upward

• Z- Score is zero if the data value is equal to the sample mean.

• #### Chebyshev’s Theorem arrow_upward

• Chebyshev's Theorem enables us to state that a proportion of data values must be within a specified number of standard deviation of the mean.
• This theorem applies to any data set regardless of the shape of the distribution of the data.

• ###### Chebyshev’s Rule:

• At least of the data values must be within z standard deviation of the mean where z is any value greater than 1.
• For z = 2, 3 and 4, the theorem states that:
• At least 75% of the observations must be contained within the distances of 2 Standard Deviation around the mean.
• At least 88.89% of the observations must be contained within the distances of 3 Standard Deviation around the mean.
• At least 93.75% of the within the distances of 4 Standard Deviation around the mean. #### Bell-Shaped Distribution arrow_upward

• “Bell curve” refers to the shape that is created when a line is plotted using the data points for an item that meets the criteria of “normal distribution”.
• • For most data sets:
• Approximately 68% of the observations fall within Standard Deviation (SD) around the mean.  • Approximately 95% of the observations fall within Standard Deviation SD around the mean.  • Approximately 99.7% of the observations fall within SD around the mean.  ##### Applications:
• Normal distributions are used in statistics to make inferences about the population mean when the sample size n is large.

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