Central Tendency Quartiles and Percentiles (الربيعيات والمئينات) 1. Quartiles: There are three Quartiles (Q1,Q2,Q3) , Q1 or first quartile which separate the25% of sorted data, Q2 or second quartile same as median and Q3 which separate the75% of sorted data. 2. Percentiles they are a numerical values of the variable that divide a set of ordered data into 100 equal parts, each set of data has 99 percentiles.

Central Tendency Quartiles and Percentiles (الربيعيات والمئينات) 1. The procedures of determines the value of kth percentiles: 2. The data must be ordered. 3. The position for percentile is founded by first calculating the value 4.The percentile it self is obtained by finding the corresponding value in ordered data.

Central Tendency Quartiles and Percentiles (الربيعيات والمئينات) Example: suppose we have the observations {7, 4, 3, 5, 6, 8, 10 ,1}, find the 30th and 50th percentiles. Sort values as:{1, 3, 4, 5, 6, 7, 8, 10} The position of 30th = 3.Then the 30th is =3.5

Central Tendency Quartiles and Percentiles (الربيعيات والمئينات) 4.The position of 50th = 5. Then the 50th is

Central Tendency Mode Mode for ungrouped data is the highest frequency. Example {1, 2, 4, 2, 2, and 4} mode here is 2.

Central Tendency For grouped data Step (1): determine the modal class (class with the highest frequency). Step (2): Calculate D1= difference between the largest frequency and frequency immediately preceding it. Step (3): Calculate D2= difference between the largest frequency and the frequency immediately following it.

Central Tendency Step (4): Calculate the mode using the following formula Where: L is the lower bound of modal class. C: is the modal class width.

Central Tendency Example: Calculate the mode from table Age Frequency 20-25 2 25-30 14 30-35 29 35-40 43 40-45 33 45-50 9 Total 130

Central Tendency Solution: Modal class is 43 L=35 D1= 43-29=14

Central Tendency Advantages of mode 1. It is more appropriate to know the most common value. 2. Easy to understand, not difficult to be calculated. 3. It is not affected by extreme values.

Central Tendency Disadvantages of Mode 1. It ignores dispersion around the mode. 2. It is unsuitable for further statistical analysis. 3. It is affected by the most popular class when a distribution is significantly skewed.

Measures of variation (dispersion) [مقاييس التشتت] Measure of variation is so important in statistics because it gives information on the spread or variability of the data values.

Measures of variation (dispersion) [مقاييس التشتت] Why study dispersion is important? A measure of location such as the mean or the median, only describes the center of the data, but it doesn’t tell us anything about the spread of the data.

Measures of variation (dispersion) [مقاييس التشتت] Example: In a hospital where each patient’s pulse rate is taken three times a day, that of a patient A is 72, 76, and 74. While that for patient B is 72, 91 and 59. The mean pulse of rate of the two patient is the same 74 but observe the difference in variability, whereas patient A’s pulse rate is stable that of patient B is not.

Measures of variation (dispersion) [مقاييس التشتت] Range The range is the difference between the highest (maximum) and lowest (minimum) observation. Range Xmax – Xmin The range can be calculated quickly, but it is not very useful.

Measures of variation (dispersion) [مقاييس التشتت] Mean deviation It is the deviation of all observation from the mean. Formula

Measures of variation (dispersion) [مقاييس التشتت] Example: the table below has a data about graduated students from national school of engineering during five years. Calculate the mean of deviation. Year of graduation 2004 2005 2006 2007 2008 No. of students 4 6 5 8 7

Measures of variation (dispersion) [مقاييس التشتت] Solution 1. Calculate mean 2. Mean Deviation = Interpretation: The mean deviation of the data about mean is equal 1.2

Measures of variation (dispersion) [مقاييس التشتت] Variance 1. The variance is a measure which uses the mean as a point of reference. 2. The variance is less when all values are close to the mean. 3.Variance is the average (approximately) of squared deviations of values from the mean.

Measures of variation (dispersion) [مقاييس التشتت] Formula: For ungrouped data population variance: 2. For ungrouped data sample variance

Measures of variation (dispersion) [مقاييس التشتت] Standard Deviation (S) 1. It is the square root of variance. 2. Most commonly used measure of variance. 3. Shows variation about mean. 4. It used to compare between more than one data set when the means are equal, the best one is the minimum.

Measures of variation (dispersion) [مقاييس التشتت] Example Calculate variance and standard deviation Year of graduation No. of Students   2004 4 -2 2005 6 2006 5 -1 1 2007 8 2 2008 7 Total 30 10  

Measures of variation (dispersion) [مقاييس التشتت] Solution: Variance Standard Deviation Interpretation: The observations fall 1.58 units from the mean.