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1- Calculation of the central tendency measures The values of a variable often tend to be concentrated around the center of the data. Some of these measures are the mean, mode and median. These measures are considered as representatives of the data. Averages give us information about a typical element of a data set. They are measures of central tendency.
(Arithmetic Mean ) The Arithmetic Mean is obtained by summing all elements of the data set and dividing by the number of elements. Example :- X1 =30 x2 = 35 x3 = 27 = 30 + 35 + 27 = 92÷3 = 30.67 n= 3
The mean is simple to calculate. Only one mean for given sample data. Only be found for quantitative variables.
Mode is the data element which occurs most frequently. Some data sets contain no repeated elements. In this case, there is no mode (or the mode is the empty set). It is also possible for two or more elements to be repeated with the same frequency. In these cases, there are two or more modes and the data set is said to be bimodal or multimodal.
-If values are different or have the same frequency there is no mode. -Data may have more than one mode. -Mode may be found for both quantitative and qualitative variables.
Example (1): The following data give the speeds (in miles per hour) of eight cars. 77- 82- 74 -81- 79- 84- 74- 78. Find the mode. miles per hour
The Median is the middle element when the data set is arranged in order of importance. The symbol (pronounced "x- tilde") is sometimes used for the median.
Example :-Find the median for the sample values :- 10, 54, 21, 38, 53 Order = arranged from the lowest to the highest number. Order set…….10, 21, 38, 53, 54
The Midrange is the arithmetic mean of the highest and lowest data elements. . Symbolically, midrange is computed as (xmax + xmin)/2
Example: A sample of size 5 (n=5) is taken of student quiz scores with the following results: 1, 7, 8, 9, 10 The midrange is (10+1)/2 = 5.5 (note the extra decimal place is required).
Dispersion Refers to how spreads out the values from each other. Measure of variation of the item. Degree of variation of the variable about a central value
Range is the difference between the highest and lowest Symbolically, range is computed as R = xmax - xmin . data element.
Examples Calculate the range from the following observation: 20, 45, 11, 47, 70 The range = xmax - xmin R = 70- 11=59
2- Variance Variance is a measurement of the spread between numbers in a data set. تانايبلا هعومجم ىف ماقرلاا نيب قرافلا سايق
5-1 We want to compute the sample variance of the following sample values:- 10, 21, 33, 53, 54. n=5 mean= 34.2 S² =(34.2- 10)²+(34.2-21)²+(34.2-33)²+(34.2 -53)²+(34.2 -54)²⁄ 4= 376.7
The Standard deviation is another way to calculate dispersion. This is the most common and useful measure because it is the average distance of each score from the mean.
Note: The SD is a measure of variation. It is the square root of the variance.
5-1 We want to compute the S.D of the following sample values:- 10, 21, 33, 53, 54. n=5 mean= 34.2 S =(34.2- 10)²+(34.2-21)²+(34.2-33)²+(34.2 - 53)²+(34.2 -54)²⁄ 4= 376.7=19.4
Definition :-Normal distribution curve is a bell -shaped curve with most scores in the middle and a small number of scores at the low and high ends.
It is the distribution that is normally seen. It is the most important tool in analysis of epidemiological and research data. The shape of the normal curve is often showed as a bell- shaped curve
The highest point on the normal curve is at the mean of the distribution
The standard deviation determines the width of the curve
It has a peak. It has two symmetric sides. The peak coincides with all central tendency measures (mean, mode, median, and midrange).
It has a peak. It has two symmetric sides. The peak coincides with all central tendency measures (mean, mode, median, and midrange). The normal curve extends in both sides
