1. Methods of mathematical presentation (Summery Statistics)
Dr. Amjad El-Shanti
MD, PMH,Dr PH
University of Palestine
2016

2. Summery Statistics Measures of Location: Measures of dispersion
Measures of central tendency
Measures of non central locations
Measures of dispersion

3. 1. Measure of central tendency
An average is a value which is representative of a set of data. Since such values tend to lie centrally within a set of data arranged according to magnitude.
Averages are called measures of central tendency.
Several types of averages can be defined, the most common being:
The midrange
The mode
The median
The mean (arithmetic, geometric, and harmonic).
Each of these averages has advantages and disadvantages or limitations depending on the data and the intended purpose.

4. Midrange I- Determination of mid range from ungrouped data:
Midrange= smallest observation+ Largest observation 2
Example: The following data represents the weight of 5 persons by kg. Find the average weight using mid-range.
(28, 30, 22, 18, 29)
Smallest observation= 18
Largest observation= 30
Mid-range= = 24kg

5. Midrange I- Determination of mid range from grouped data:
Midrange= lower limit of first interval+ upper limit of last interval 2
Example: For quantitative continuous variable
Lower limit of first interval= 25
Upper limit of last interval= 75
Mid-range= = 50 years
Age in Years
Frequency
25- 5
30- 2
35- 14
40- 9
60-75 4
Total 34

6. Midrange II- Determination of mid range from grouped data:
Midrange= lower limit of first interval+ upper limit of last interval 2
Example: For quantitative discrete variable
Lower limit of first interval= 1
Upper limit of last interval= 20
Mid-range= = 10.5 cigarettes
Number of cigarettes per day
Frequency 1- 10 5- 20
10- 5
15-20
Total 40

7. Midrange
Advantages:
The midrange is easily and quickly obtained and therefore it is used only when we want to obtain a rough and quick idea about the average of a set of observations.
Disadvantages:
The mid range is a rough measure of central tendency since it neglects all intermediate observations.
The midrange is seriously affected by extreme or outlying observations.
The midrange can b used only with quantitative variables. It can not be used with qualitative variables.
The mid range can not be computed for open ended tables.
The mid range can not be used in statistical analysis.

8. The Mode
The mode of a set of numbers is that value which occurs with the greatest frequency. i.e. It is the most common value.
The mode may not exist, and even if it does exist it may not be unique.
I- Determination of the mode from ungrouped data:
Quantitative :
Example 1: The set 2,2,5,7,9,9,9,10,10,11,12,18 has mode 9 and is called unimodal.
Example 2: The set 2,5,8,10,12,15,16 has no mode.
Example 3: The set 2,3,4,4,4,5,5,7,7,7,9 has 2 modes 4 and 7 and is called bimodal.
Qualitative :
Example 1: The set of blood groups :A,B,A,AB, O,B,A has mode A. (unimodal)
Example 2: The set of educational level : primary, secondary, university, illiterate, read and right, preparatory has no mode.
Example 3: The set of blood group B,O,AB,A,O,AB,AB,O has 2 modes: O, andAB (bimodal).

9. The Mode There are 4 methods: Method of modal interval
II- Determination of the mode from grouped data:
For Quantitative variables:
There are 4 methods:
Method of modal interval
Mid point of modal interval
Graphical method
Lever method

10. The Mode 1. Method of modal interval
II- Determination of the mode from grouped data:
For Quantitative variables:
1. Method of modal interval
The modal interval is the class interval with the highest frequency.
Example:
The modal interval = years
Age in years
Frequency
25- 5
30- 2
35- 14
40- 9
60-74 4
Total 34

11. The Mode 2. Mid point of modal interval
II- Determination of the mode from grouped data:
For Quantitative variables:
2. Mid point of modal interval
Mid point= Lower limit of modal interval +upper limit of modal interval 2
Example:
Mid point= = years
Age in years
Frequency
25- 5
30- 2
35- 14
40- 9
60-74 4
Total 34

12. The Mode 2. Mid point of modal interval
II- Determination of the mode from grouped data:
For Quantitative variables:
2. Mid point of modal interval
Example: Quantitative discrete variables
Modal interval =5-9 cigarettes
Mid point= 5+9= 7 cigarettes 2
Number of cigarettes
Frequency 1- 10 5- 20
10- 5
15-20
Total 40

13. The Mode 3. Graphical method
II- Determination of the mode from grouped data:
For Quantitative variables:
3. Graphical method
This can be used only for determination of mode for quantitative continuous variables because this method requires drawing a histogram in order to determine the mode.
Steps:
1- Draw a histogram from the simple frequency distribution table.
2- Identify modal, pre-modal and post-modal bars. (Modal bar= longest bar).
3- Join the upper right hand corner of pre-modal bar with the upper right hand corner of the modal bar by a straight line.
4- Join the upper left hand corner of post-modal bar with the upper left hand corner of the modal bar by a straight line.
5- From the point of intersection, drop a vertical line on the horizontal axis and read the value of the mode.

14. The Mode 3. Graphical method
II- Determination of the mode from grouped data:
For Quantitative variables:
3. Graphical method
Example:
Age (Years)
Frequency 5- 3
10- 5
15- 7
20-24 2
Total 17

15. Force X length (X)= Resistance X (total length-X)
The Mode
II- Determination of the mode from grouped data:
For Quantitative variables:
4. Lever method
Used for quantitative continuous discrete data.
Steps:
1- We assume that the modal interval is represented by a lever of the first type and that the total length of the lever is equal to the length of the modal interval.
We also assume that at one end the lever is affected by a force equal to the pre-modal frequency and at the other end a resistance equal to the post-modal frequency.
We also assume that the distance between the force and the center of gravity is equal to x and hence the distance from resistance to that center will be equal to (total length of the lever-x)
2- We solve for the value of x using the law:
Force X length (X)= Resistance X (total length-X)
3- Determine the mode by adding the value of X to the lower limit of the modal interval.

16. The Mode
II- Determination of the mode from grouped data:
For Quantitative variables:
4. Lever method
Example:
Modal interval=
Length of modal interval= 5
8(X)= 6(5-x)
8X =30-6X
14X= 30
X=30=2.1 14
Mode= =57.1kg
Resistance Force
5-X
Weight (Kg)
Frequency
40- 3
45- 5
50- 8
55- 12
60- 6
65-69
Total 37

17. The Mode II- Determination of the mode from grouped data:
For Qualitative variables:
In this case there is only one method for determining the mode and this is by determining the category with the highest frequency
In the previous example is the blood group B.
Blood Group
Frequency A 10 AB 14 B 25 O 9
Total 58

18. The Mode
Advantages:
1- The mode can be used for all types of variables.
2- It is easy to determine
3- It is never affected by extreme or outlying observation (since by definition the mode is the most frequent observation and the extreme is the most rare one).
4- The mode could be obtained from closed ended or open ended tables.
Disadvantages:
1- The mode neglects all the less frequent observations.
2- Sometimes we may have 2 modes (Bimodal )or more than two mode (Multi-modal) in the same set of observations.
3-Sometimes the mode can not be determined, this happens when all the observations have the same frequency.
4- The mode is not easily used in statistical analysis.

19. The Median
The median is defined as the observation which lies in the middle of the ordered observation.
I- Determination of the median from ungrouped data:
For Quantitative Data :
When the number of observation is odd:
1-Arrange the observation in an ascending or descending order
(Usually for ease of calculation we use ascending order).
2- determine the rank of the median which is given by:
Rank of median= n+1 ,Where n= number of observations (sample size) 2
3- Using the obtained rank and referring back to the ordered observations to determine the value of median.
Example: 24, 18, 22,16, 20 kg
N= Odd
1- arrange in ascending order:
2- Rank of median 5+1= 3
3- Median= 20 kg.

20. The Median I- Determination of the median from ungrouped data:
For Quantitative Data :
B) When the number of observation is even:
1-Arrange the observation in an ascending or descending order
2- determine the ranks of the two middle observation which are given by:
Rank s= n , n +1
3- Using the obtained rank and referring back to the ordered observations to determine the two middle values and compute the value of the median as follows:
Median= Sum of two middle observations 2
Example: 26, 24, 18, 22,16, 20 kg
N= even
1- arrange in ascending order: kg
2- Rank of two middle observations:
1st rank= n/2= 6/2 = , 2nd rank = n +1 = 3+1 = 4
3- Median= =42 = 21 kg

21. The Median I- Determination of the median from ungrouped data:
For Qualitative Data :
When the number of observation is odd: .
Example: Level of education
Secondary, University, primary, preparatory, University.
1- arrange in ascending order:
Primary, Preparatory, Secondary, University, University
2- Rank of median 5+1= 3 2
3- Median= Secondary level of education.

22. The Median I- Determination of the median from ungrouped data:
For Qualitative Data :
B) When the number of observation is even:
Example: Grades of success
Very good, good, weak, excellent, good, good, excellent, excellent.
1- arrange in ascending order:
Weak, good, good, good, very good, excellent, excellent, excellent.
2- Ranks of two middle observations
1st rank= n/2= 8/2 = , 2nd rank = n +1 = 4+1 = 5 2
3- Median= Good, very good.

23. The Median II- Determination of the median from grouped data: Example:
Weight (Kg)
Observed frequency
Ascending Cumulative Frequency
40- 3
45- 4 7
50- 8 15
55- 5 20
60-64 24
Total

24. The Median II- Determination of the median from grouped data:
The steps for finding the median for grouped data are as follows:
1- Find the ascending cumulative frequency value for each category in the table which is defined as the number of observation whose values are less than the upper limit of the interval and note that:
a) The ascending cumulative frequency (A.C.F.) of first interval is always equal to the observed frequency of that interval.
b) The A.C.F. of the last interval is always equal to the sum or total of the observed frequency.
c) The A.C.F. Values should not added together.
2- Determine the general rank of the median which is equal to ∑f/2 irrespective of whether n id odd or even and where n is equal the sum of observed frequencies.
3- Determine the median interval by comparing the general rank of the median with the A.C.F. values from above down-wards and first interval met with whose A.C.F. is equal to or greater than the general rank of the median is the median interval.
4- Determine the special rank of the median given by:
Special rank = General rank – A.C.F of pre-median interval.
5- Determine the value of the median given by:
Median = Lower limit of median + (Special rank X width of median interval )
observed frequency of
median interval

25. The Median II- Determination of the median from grouped data:
Solution of Example:
Find the ascending cumulative frequency.
General rank= ∑ f/2= 24/2= 12
By comparison with A.C.F. so median interval=
Special rank of median= 12-7= 5
Median= X 5/8
= /8 =
= kg.

26. The Median Advantages:
1- The median can be used wit 3 types of variables namely quantitative continuous, quantitative discrete and qualitative ordinal.
2- It is the best measure of central tendency for skewed distribution.
3- The median could be computed whether the table is opened or closed.
4- The median is not affected by extreme or outlying observations that is why the median is the average of choice when dealing with certain observations in which we usually meet extreme or outlying values such as bacteriology, virology, biological assay and toxicology .
Disadvantages:
1- One disadvantage for the median is that it does not take all observations into consideration.
2- The median is not easy to deal with in statistical analysis.

27. The Arithmetic Mean
Computation of the arithmetic mean from ungrouped data:
By definition the arithmetic mean of a set of observations is given by:
Arithmetic mean= Sum of all observations
Number of observations
Example:
The following observation represent the weight of 5 persons 24, 20, 22, 16, 18 kg
Determine the arithmetic mean?
Solution:
Arithmetic mean = = 100 = 20 kg

28. The Arithmetic Mean
Computation of the arithmetic mean from ungrouped data:
In general any observations can be denoted by X where the subscript (i) will take numerical values depending on the order of appearance of the observation for example i= 1 for the first observation, i= 2 for the second observation and so on until i= n for the last observation where n= total number of observations. The Greek letter ∑ implies a summation operation. The limits of the summation operation can be shown below and above the letter ∑. The arithmetic mean itself is denoted by X and is read (X bar).
In the previous example
X1= 24 1st observation
X2= 20 2nd observation
X3= 22 3rd observation
X4= 16 4th observation
X5= 18 5th observation
So Xi= ith observation
Where i= 1,2,3,4 or 5.
Arithmetic mean (x)= 5
The formula to obtain the arithmetic mean from the ungrouped data will be: ∑ xi
X= i=1 n
Where n = number of observations.

29. The Arithmetic Mean
B) Computation of the arithmetic mean from grouped data using the general or long method:
1- Determine the mid-point of each interval (Xj)
Xj = Lower limit + Upper limit 2
If the variable is of the continuous type:
Upper limit of interval= Lower limit of next interval.
If the variable is of the discrete type:
Upper limit of interval = Lower limit of next interval
2- Find the product fi xi for each interval. c
3- Find the sum of the above products . ∑ fj xi
where c= total number of categories in the table.
4- Find the arithmetic mean from the following formula:
∑ f j Xi
X= j=1
∑ f j

30. Mid-point of interval Xj
The Arithmetic Mean
Computation of the arithmetic mean from grouped data using the general or long method:
Example : quantitative continuous variable:
X= 720 = 36 kg 20
Weight (kg)
Frequency Fj
Mid-point of interval Xj
Fj Xi
15- 3 20 60
25- 6 30
180
35- 8 40
320
45- 2 50
100
55-64 1
Total
720

31. Mid-point of interval Xj
The Arithmetic Mean
Computation of the arithmetic mean from grouped data using the general or long method:
Example : quantitative Discrete variable:
X= 78 = 4.6 rooms 17
Number of rooms
Frequency Fj
Mid-point of interval Xj
Fj Xi 1- 5
1.5
7.5 3- 4
3.5 14 5- 3
5.5
16.5
7-9 8 40
Total 17 78

32. The Arithmetic Mean Advantages:
1-It is considered the best type of average for quantitative variables, because it takes all observations into consideration.
2- It is easily used in statistical analysis.
Disadvantages:
1- It can not be used with qualitative variables.
2- It is affected by extreme or outlying observations.
3- It can not be computed from open-ended tables.

33. The Arithmetic Mean Example:
The following are the plasma volumes of eight healthy adult males:
– 2.76 – 2.62 – 3.49 – 3.05 – 3.12 liters
-Calculate possible measures of central tendency.
Solution:
Arrange data: – 3.05 – 3.12 – 3.37 – 3.49
Mid range = = 3.06 liters 2
Mode There is no estimate of the mode, since all the values are different.
Median even
1st rank = n/2= 4
2nd rank = (n/2)+1= 5
Median = the average of the 4th and 5th observations
= ( )/2 = 2.96 liters
Mean= X= ∑ xi /n = 24.02/8 = 3 liters
= 24.02