Outcome: The student by the end of the lecture will be acquainted with The different measures of central tendency and when to use each and how they are calculated.

Mathematical presentation: Data are summarized into one or more measures which are obtained through certain calculations or certain mathematical formulae. This group is further subdivided into two subgroups:  Measures of central tendency (Averages).  Measures of dispersion.

Measures of central tendency Advantages: it is quickly and easily obtained. Disadvantage: it is used only with quantitative data and cannot be used with qualitative ones. Also it dose not take into consideration except the smallest and the largest observations neglecting all intermediate observations, hence the midrange is useful as a method to obtain a quick but rough average for a group of data.

The Mode: Example: 3,5,7,7,9. The mode is defined as the observation with the highest frequency. It has the advantage of being used with any kind of variable whether quantitative or qualitative, yet it has the following disadvantages: 1. Sometimes the mode cannot be determined, this happens if all the observations are repeated the same number of times i.e: they have equal frequencies for e.g. 3,5,7,9. 2. Sometimes a group of data may have two modes in which case it is bimodal or more than two modes i.e. multimodal 3,5,7,7,9,9.

The Arithmetic mean: Any observation is denoted by where the “I” is a subscript that take the value from 1 up to “n” where n is the total number of observations. The mean is denoted by and it can be obtained from the formula: The limits of the summation are shown below and above the letter “∑” which denotes the summation operation.

Disadvantages of the mean: It is only used for quantitative data It is greatly affected by extreme observations Ex: if the blood pressure of five persons is 110,115,130,210 It is obvious that if we describe this group of data as having a mean blood pressure of 137 such description may be misleading since the majority of the group have a much lower B.P and the above result was greatly affected by the single extreme observation of 210

Computation of the mean from grouped data: Using the long method: 410 is the estimated sum of all observations Fj xjMid point “xj” of intervalFrequency “fj”Age in years Total

Steps: Determine xj the mid point for each interval which is obtained by It should be noted that the upper limit of the interval will be equal to the lower limit of the next interval in the case of the continuous variable while in the case of discrete variables the upper limit is one less than the lower limit of the next interval.

The Median The value which have 50% of the data above and 50% of the data below it

The median from ungrouped observations: To get the median arrange the values into an ascending order so that an equal no of observations lie on either side of it eg. If the grades of 7 student are: out of 10 The value of the median will be: Arrange data into an ascending order: then get the order of the median by In case of odd number of observations the rank of the median is So the median will be 6.

If the number of observations is even the first rank of the median will be and the second rank be. The median from grouped data: to get the value of the median arrange data in an ascending order, then count half the frequency by using eg. Suppose the heights of 120 males as follows:

To determine the rank of median ASC.C.F.FrequencyHight TOTAL

Get the ascending commutative frequency by adding the class frequency from the lowest class till you reach the highest one. So the median will be some where above 170cm (as we only have 50 students up to 170)& below 180, in other words we need only 10 individuals out of the 45 of the 3 rd class to reach position of the median. The value of the median would be 10/45 of the 3 rd class interval added ot its beginning, that is the median = (Rank of median – a.c.f. pre median interval)

Questions: Write the equation for calculation of each of the following 1. Mid range 2. The mode 3. The median 4. The auiltmatic mean

TitleStudent Name Graphical presentation versus mathematical presentation. احمد شيخون محفوظ احمد محمد احمد عبد العال احمد محمد فوزي اخلاص حمدسي اسراء علي اسلام عادل اسلام محمد عبد السلام اسماء ابراهيم كمال اسماء النقراشي الشيماء مصطفي عبد العاطي

References Biostatistical analysis: Jerrold H. Zar