Chebychev’s theorem:
For any number k greater than 1, at least 1 – 1/k2 of the data-values fall within k standard deviations of the mean, i.e., within the interval (`X – kS,`X + kS)
This means that:
a) At least 1-1/22 = 3/4 will fall within 2 standard deviations of the mean, i.e. within the interval
(`X – 2S,`X + 2S).
b) At least 1-1/32=8/9 of the data-values will fall within 3 standard deviations of the mean, i.e. within the interval (`X – 3S,`X + 3S)
Because of the fact that Chebychev’s theorem requires k to be greater than 1, therefore no useful information is provided by this theorem on the fraction of measurements that fall within 1 standard deviation of the mean, i.e. within the interval (X–S,`X+S).
Next, let us consider the Empirical Rule mentioned above.
FIVE-NUMBER SUMMARY | |
A five-number summary consists of X0,Q1, Median, Q3, and XmIt provides us quite a good idea about the shape of the distribution.If the data were perfectly symmetrical, the following would be true:1. The distance from Q1 to the median would be equal to the distance from the median to Q3:THE SYMMETRIC CURVE. The distance from X0 to Q1 would be equal to the distance from Q3 to Xm. 3. The median, the mid-quartile range, and the midrange would all be equal. All these measures would also be equal to the arithmetic mean of the data:On the other hand, for non-symmetrical distributions, the following would be true:1. In right-skewed distributions the distance from Q3 to Xm greatly exceeds the distance from X0 to Q1. 2. in right-skewed distributions, median < mid-quartile range < midrange:Similarly, in left-skewed distributions, the distance from X0 to Q1 greatly exceeds the distance from Q3 to Xm.Also, in left-skewed distributions, midrange < mid-quartile range < median.Let us try to understand this concept with the help of an example EXAMPLE: Suppose that a study is being conducted regarding the annual costs incurred by students attending public versus private colleges and universities in the United States of America. In particular, suppose, for exploratory purposes, our sample consists of 10 Universities whose athletic programs are members of the ‘Big Ten’ Conference. The annual costs incurred for tuition fees, room, and board at 10 schools belonging to Big Ten Conference are given as follows:If we wish to state the five-number summary for these data, the first step will be to arrange our data-set in ascending order:Ordered Array:And if we carry out the relevant computations, we find that:MEDIANAND QUARTILES
The Essay on Team Data Collection
BIMS, Ballard Integrated Managed Services, Inc. is a corporation specializes in the services of food and housekeeping to corporations and institutions. The general manager, Barbara Tucker, has noticed that over the last four months the turnover rate has shot up to 64%. Not only is the turnover rate higher than usual, but also employees seemed unmotivated to do their job. There are more employees ...
FOR THIS DATA-SET:1) The median for this data comes out to be 15.30 thousand dollars. 2) The first quartile comes out to be 14.90 thousand dollars, and 3) The third quartile comes out to be 16.40 thousand dollars. Therefore, the five-number summary for this data-set is: The Five-Number Summary: If we apply the rules that I am conveyed to you a short while ago, it is clear that the annual cost data for our sample are right-skewed. We come to this conclusion because of two reasons:1. The distance from Q3 to Xm (i.e., 6.7) greatly exceeds the distance from X0 to Q1 (i.e., 1.9).
2. If we compare the median (which is 15.3), the mid-quartile range (which is 15.65), and the midrange (which is 18.05), we observe that the median < the mid-quartile range < the midrange.Both these points clearly indicate that our distribution is positively skewed.The gist of the above discussion is that the five-number summary is a simple yet effective way of determining the shape of our frequency distribution — without actually drawing the graph of the frequency distribution |
The Essay on Probability Distribution
In the world of statistics, we are introduced to the concept of probability. On page 146 of our text, it defines probability as “a value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur” (Lind, 2012). When we think about how much this concept pops up within our daily lives, we might be shocked to find the results. ...
Lecture no 13
Box and whisker plots.
Coefficient of skewness.
Lesson no 14
• Bowley’s coefficient of skewness• The Concept of Kurtosis• Percentile Coefficient of Kurtosis• Moments & Moment Ratios• Sheppard’s Corrections• The Role of Moments in Describing Frequency Distributions |
Pearson’s coefficient of skewness:
Lesson 15
• Simple Linear Regression
• Standard Error of Estimate
• Correlation
Introduction | |
• Permutations• Combinations• Random Experiment• Sample Space • Events• Mutually Exclusive Events• Exhaustive Events• Equally Likely Events |
• Subjective Approach to Probability
• Objective Approach
• Classical Definition of Probability
Introduction | |
• Relative Frequency Definition of Probability• Axiomatic Definition of Probability• Laws of Probability• Rule of Complementation• Addition Theorem |
Application of Addition Theorem
• Conditional Probability
• Multiplication Theorem
Independent and Dependent Events
• Multiplication Theorem of Probability for Independent Events
• Marginal Probability
Bayes’ Theorem
Discrete Random Variable
Discrete Probability Distribution
Graphical Representation of a Discrete Probability Distribution
Mean, Standard Deviation and Coefficient of Variation of a Discrete Probability Distribution
Distribution Function of a Discrete Random Variable.