## Archive for August, 2014

## Standard Deviation for a grouped distribution

**Standard Deviation for a grouped distribution**

The standard deviation for a distribution with mean , where x is an interval midpoint with frequency f, and n = ∑f , is

s = S = .

## Area Under a Normal Curve

Area Under a Normal Curve

The area under a normal curve between x = a and

x = b is the same as the area under the standard normal curve between

the z – score for a and the z – score for b.

## Mean and Standard Deviation for the Binomial Distribution

**Mean and Standard Deviation for the Binomial Distribution**

For the binomial distribution, the mean and standard deviation are given by

µ = np and

where n is the number of trials and p is the probability of success on a single trial.

## z- Score

**z- Score**

If a normal distribution has mean µ and standard deviation , then the z-score for the number x is

z = .

Example:- If a normal distribution has mean 50 and standard deviation 4, find the following

*z*-scores. The *z*-score for x= 46

**Solution :- **Since 46 is 4 units below 50 and the standard deviation is 4, 46 is

1 standard deviation below the mean. So, its *z*-score is -1.

## Standard Deviation

**Standard Deviation**

The standard deviation of the n numbers x, x, ,………..x, with mean , is

S = .

## Variance

**Variance**

The variance of a sample of n numbers x_{1}, x_{2},……..x_{n}, which mean , is

S^{2} =

## Deviations from the Mean

**Deviations from the Mean**

** **

Find the deviations from the mean for the numbers

32, 41, 47, 53, 57.

Solution:- Adding these numbers and dividing by 5 given a mean of 46.

To find the deviations from the mean, subtract 46 from each numbers in the list. For example, the first deviation from the mean is 32 – 46 = -14; the last is 57 – 46 = 11.

To check your work, find the sum of these deviations. It should always equal 0.

The answer is always 0 because the positive and negative numbers cancel each other.

## Range

**Range**

The difference between the largest and smallest number in a sample is called the range.

Example:-

Find the range for each list numbers.

12, 27, 6, 19, 38, 9, 42, 15

Solution:-

The highest number here is 42; the lowest is 6. The range is the difference between these numbers, or

42 – 6 = 36.

## Mode

**Mode**

The number occurs more often then any other, it is called mode.

Example:-

57, 38, 55, 80, 55, 87, 98

Solution:- The number 55 occurs more often then any other, so it is the mode. It is not necessary to place the numbers in numerical order when looking for the mode.

## Median

Median

**Odd Number of Entries**

8, 7, 4, 3, 1

The median is a middle term “4”.

** **

**Even Number of Entries**

2, 3, 4, 7, 9, 12

The median is average value of two middle terms

= 5.5