Rules of Data Dispersion

• By using the mean and standard deviation, we can find the percentage of total observations that fall within the given interval about the mean.

Rules of Data Dispersion

• Empirical Rule• Chebyshev’s Theorem

(IMPORTANT TERM: AT LEAST)

Empirical Rule

Applicable for a symmetric bell shaped distribution / normal distribution.There are 3 rules:i. 68% of the observations lie in the interval

(mean ±SD)ii. 95% of the observations lie in the interval

(mean ±2SD)iii. 99.7% of the observations lie in the interval

(mean ±3SD)

Empirical Rule

Empirical Rule

• Example: 95% of students at school are between 1.1m and 1.7m tall. Assuming this data is normally distributed can you calculate the mean and standard deviation?

Empirical Rule

Empirical Rule

• The age distribution of a sample of 5000 persons is bell shaped with a mean of 40 yrs and a standard deviation of 12 yrs. Determine the approximate percentage of people who are 16 to 64 yrs old.

Chebyshev’s Theorem

Chebyshev’s Theorem

• Applicable for any distribution /not normal distribution

• At least of the observations will be in the range of k standard deviation from mean where k is the positive number exceed 1 or (k>1).

2

1(1 )

k

Chebyshev’s Theorem

• Example Assuming that the weight of students in this

class are not normally distributed, find the percentage of student that falls under 2SD.

Chebyshev’s Theorem

• Consider a distribution of test scores that are badly skewed to the right, with a sample mean of 80 and a sample standard deviation of 5. If k=2, what is the percentage of the data fall in the interval from mean?

Measures of Position

To describe the relative position of a certain data value within the entire set of data.•z scores•Percentiles•Quartiles•Outliers

Quartiles

• Divide data sets into fourths or four equal parts.

Boxplot

1

3

Lower Fence 1.5( )

Upper Fence 1.5( )

Q IQR

Q IQR

3 1IQR Q Q

Boxplot

Outliers

• Extreme observations• Can occur because of the error in

measurement of a variable, during data entry or errors in sampling.

Outliers

Checking for outliers by using Quartiles

Step 1: Determine the first and third quartiles of data.Step 2: Compute the interquartile range (IQR).

Step 3: Determine the fences. Fences serve as cutoff points for determining outliers.

Step 4: If data value is less than the lower fence or greater than the upper fence, considered outlier.

3 1IQR Q Q

1

3

Lower Fence 1.5( )

Upper Fence 1.5( )

Q IQR

Q IQR