Measures of Variability OBJECTIVES To understand the different measures of variability To determine the range, variance, quartile deviation, mean deviation and standard deviation for ungrouped and grouped data

Measures of dispersion (variability or spread) consider the extent to which the observations vary

MEASURES OF VARIATION RANGE QUARTILE DEVIATION MEAN DEVIATION VARIANCE STANDARD DEVIATION

1. Range, R The difference in value between the highest-valued data, H, and the lowest- valued data, L R = H – L Example: 3, 3, 5, 6, 8 R = H – L = 8 – 3 = 5

2. Quartile Deviation, QD or semi-interquartile range obtained by getting one half the difference between the third and the first quartiles

SOLVE FOR Q 1 and Q 3

Problem The examination scores of 50 students in a statistics class resulted to the following values: Q 3 = Q 1 = Determine the value of the quartile deviation or semi-interquartile range.

Solution

Problem Compute the value of the semi- inter quartile range or quartile deviation The performance ratings of 100 faculty members of a certain college are presented in a frequency distribution as follows:

Class interval or Classes f <cf st quartile class rd quartile class

Solution (Grouped data)

Solution cont’d…

3. Mean Deviation, MD – based on all items in a distribution

For ungrouped data For grouped data

4. Variance, s 2 - most commonly used measure of variability - the square of standard deviation

For ungrouped data

Note: The greater the variability of the observations in a data set, the greater the variance. If there is no variability of the observations, that is, if all are equal and hence, all are equal to the mean then s 2 = 0

For grouped data

5. Standard Deviation, s - the positive square root of the variance

Problem: Find the (a) range, (b) quartile deviation, © mean deviation, (d) variance and (e) standard deviation Student Score lowest value highest value

(a) Range, R R = H – L R = 73 – 48 = 25

(b) Quartile Deviation, QD Arrangement in ascending order Using method 3 for finding Q n (ungrouped data) Q 1 is located at n/4 = 10/4 = 2.5 Q 1 = (50+60)/2 = 55 Q 3 is located at 3n/4 = 3(10)/4 = 7.5 Q 3 = (67+71)/2 =69

QD cont’d…

© Mean Deviation, MD

First, solve for the mean Ungrouped data

Data for mean deviation, MD Score, xx i = x- x i TOTAL  x i  =

(d) Variance, s 2

(e) Standard Deviation, s

Problem: The following are marks obtained by a group of 40 students on an English examination Classesf<cf f

Find the following: a. range b. quartile deviation c. mean deviation d. variance e. standard deviation

Solution a. Range, R = H – L = 99 – 35 = 64

b. Quartile Deviation, QD

Solve for Q 1

Substitute

c. Mean Deviation, MD

Data for mean deviation, MD Class intervalxffx|x i |f|x i | Total

d. Variance, s 2

Data for the variance, s 2 Class intervalxffxxixi fx i Total409660

e. Standard Deviation, s

New Topic…

Objectives To know the measures of skewness and kurtosis To find the Pearsonian coefficient of skewness

Measures of Skewness summarize the extent to which the observations are symmetrically distributed

Skewness the degree to which a distribution departs from symmetry about its mean value or refers to asymmetry (or "tapering") in the distribution of sample data

Positive skew the right tail is longer the mass of the distribution is concentrated on the left of the figure has a few relatively high values the distribution is said to be right-skewed mean > median > mode the skewness is greater than zero

Negative skew the left tail is longer the mass of the distribution is concentrated on the right of the figure has a few relatively low values the distribution is said to be left-skewed mean < median < mode the skewness is lower than zero

No skew the distribution is symmetric like the bell-shaped normal curve bell-shaped normal curve mean = median = mode

OR…

Exercise

Pearsonian coefficient of skewness

Skewness based on quartiles

Interpretation If skewness is positive, the data are positively skewed or skewed right, meaning that the right tail of the distribution is longer than the left. If skewness is negative, the data are negatively skewed or skewed left, meaning that the left tail is longer.

Interpretation cont’d… If skewness = 0, the data are perfectly symmetrical. But a skewness of exactly zero is quite unlikely for real-world data, so how can you interpret the skewness number? In the classic Principles of Statistics (1965), M.G. Bulmer suggests this rule of thumb:

Interpretation cont’d… If skewness is less than −1 or greater than +1, the distribution is highly skewed. If skewness is between −1 and −½ or between +½ and +1, the distribution is moderately skewed.

Interpretation cont’d… If skewness is between −½ and +½, the distribution is approximately symmetric. Example: With a skewness of −0.1082, the sample data are approximately symmetric.

Problem Find the Pearsonian coefficient of skewness of the set of data shown in the following table:

Scores of ten students in a mathematics ability test Student Score

Computed values Refer to the previous computations

Interpretation Negative sign means the tail extends to the left the mean is less than the mode by 176% considered a substantial departure from symmetry

Problem Find the Pearsonian coefficient of skewness for the following set of data:

Class intervalxffx|x i |f|x i | Total

Interpretation Negative (-) computed value means the mean is less than the mode by 76.1% considered quite negligible departure from symmetry given set of data is more or less evenly distributed

Problem Find the Pearsonian coefficient of skewness for the distribution whose

Solution

Interpretation Positive sign indicates the tail of the distribution extends to the right Computed value means the mean is greater than the mode by 38% considered negligible skewness

Measures of Kurtosis Kurtosis - the degree of peakedness (or flatness) of a distribution

Types of Kurtosis Mesokurtic distribution a normal distribution, neither too peaked nor too flat its kurtosis (Ku) is equal to 3

Leptokurtic distribution has a higher peak than the normal distribution with narrow humps and heavier tails its kurtosis (Ku) is higher than 3

Platykurtic distribution has a lower peak than a normal distribution flat distributions with values evenly distributed about the center with broad humps and short tails its kurtosis (Ku) is less than 3