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 x1, x2,……..xn, which mean , is

S2 =

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