Download Statistics Summary 4: Measures of Central Tendency Descriptive

Statistics
Summary 4: Measures of Central Tendency
Descriptive measures of a distribution: they are numerical values which are used to describe
certain characteristics of a group of data
If the numerical values describe the whole population, they are called parameters. To represent
parameters Greek letters are generally used.
If the numerical values describe a sample, they are called statistics of the sample.
I. Measures of central tendency: mean, median and mode
II. Measure of variability, dispersion or spread: range, variance and standard deviation
III. Measures of relative standing: percentiles, quartiles, deciles and z-scores.
IV. Measures of position: mean, median, percentiles, quartiles and deciles.
V. Measures of skewness: symmetric, positively skewed, negatively skewed.
VI. Measures of peakness: platykurtic, mesokurtic and leptokertic.
I. Measures of central tendency:
Mean: the mean is also called the average or the center of gravity.

The average of a variable x in a sample is represented by x 
x
n
where x is the number
of observations in the sample

The average of a variable x in a population is represented by the parameter  
x
N
where N is the size of the population.
How to find the mean in the case of grouped data: in grouped data the mean can be found or
 xf where x is the class mark and f is the frequency of the
estimated by the formula x 
n
class.
Example: Estimate the mean of a sample that has been grouped according to the following
distribution:
Class
15-24
25-34
35-44
45-54
55-65
66-74
Sum
x
freque
ncy
f
8
12
13
23
14
12
82
Class Mark
x
19.5
29.5
39.5
49.5
59.5
69.5
xf
156
354
513.5
1138.5
833
834
3829
3829
 46.70
82
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The mean of means (grand-mean)- to find the average of means x1 , x2 , ... , xk , use the formula
x
n1 x1  n2 x2  ... nk xk
n1  n2  ...  nk
Problem: the average of the 17 females in a Statistics class is 83 points and the average of the
14 males was 78. What is the average of the whole class?
n x  n2 x2 17(83)  14(78) 2503
Solution: x  1 1


 80.74
n1  n2
17  14
31
The median: the median is the value at the middle after they are arranged in either ascending or
descending order.
To find the median in the case of ungrouped data:
Step 1. arrange the values in order.
Step 2. If n is odd, the middle value is the median
If n is even, the average of the two middle values is the median
Problem: find the median of the values 8, 5, 7, 13, 2, 13, 9
Step 1: 2, 5, 7, 8, 9, 13, 13
Step 2: the median is the middle value 8.
Problem: find the median of the values 8, 5, 7, 13, 2, 13, 9, 14
Step 1: 2, 5, 7, 8, 9, 13, 13, 14
Step 2: the median is the average value 8 and 9 which is 8.5
Mode: the mode, if any, is the most frequent value.
A distribution could have only one mode (unimodal), could have two modes (bimodal),
three modes (tri-modal) or many modes (multimodal)
Note: in a distribution of classes, the most frequent class is called the modal class. The mode is
given by the class mark of the modal class.
Problem: in the distribution below the modal class is the fourth class, 45-54 and the mode is
given by the class mark which is 49.5
frequency
Class
f
15-24
8
25-34
12
35-44
13
45-54
23
55-65
14
66-74
12
Characteristics:
1. The mean is very sensitive to extreme values, that is, it tends to go in the direction of extreme
values. The mean is preferred in the case of numerical data.
2. The median is not sensitive to extreme values. It is preferred in the case of ordinal data.
3. The mode is mostly used with nominal data.
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