1. STATISTIC : DESCRIPTIVE MEASURES
Measures For Central Tendency
Mean, Mode, Median ungrouped data
Mean _
x = ∑x n
Where n is the sample
Mean
µ = ∑x N
Where N is the population

2. STATISTIC : DESCRIPTIVE MEASURES
Example : Mean
The number of 911 calls classified as domestic disturbance calls in large metropolitan location where sampled for 10 randomly selected 24 hour periods with the following results. Find the mean number of calls per
24 hours period
Mean _
x =
∑250 10 = 25

3. STATISTIC : DESCRIPTIVE MEASURES
Median
Median of a set of data is a value that divides the bottom 50% of the data from
the top 50% of the data. To find the median of a data set, first arrange the data in
increasing order. If the number of observations is odd then the median is the
Number in the middle of the observation list. If the number is even then the median
Is the mean of the two values closest to the middle of the ordered list ~ ~ x
= Sample median
= Population median
Example : Median
The number of 911 calls classified as domestic disturbance calls in large metropolitan location where sampled for 10 randomly selected 24 hour periods with the following results. Find the median number of calls per
24 hours period
(even number of observation) ~ x = 2 20

4. STATISTIC : DESCRIPTIVE MEASURES
Mode
The mode is the value in a data set that occurs the most often. If no such value
exists, we say that the data has no mode. If two such values exist, we say the
data is bimodal. If three such values exist we say the data set is trimodal. There
is no symbol that is used to represent the mode.
Data set : 10, 12, 15, 15, 18, 20 Mode: 15
Shapes of Distribution
(i) Bell-shaped
(ii) Left-skewed
(iii) Right-skewed
Bell-shaped

5. STATISTIC : DESCRIPTIVE MEASURES
Shapes of Distribution
Left-skewed
Right-skewed

6. STATISTIC : DESCRIPTIVE MEASURES
Measures For Central Tendency
Mean, Mode, Median grouped data
Age
Frequency
Class Marks
Class width
5-14 7
9.5 10
15-24 15
19.5
25-34 5
29.5
35-44
39.5
45-54
49.5
Mean = x =
∑xf n
= where x represent the class marks, f represent the
frequencies and n represent the sample size
Mean = x = 9.5 x x x x x 5 37

7. Mode
Spread evenly the number of size, size is 37 so the middle value should be 19
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37
Median size
Age
Frequency
Class Marks
Class width
5-14 7
9.5 10
15-24 15
19.5
25-34 5
29.5
35-44
39.5
45-54
49.5
According to the table the median size would be at the class range
To be able to reach the median size we need to add 12 to the upper frequency
Values from the above class range ( to reach 19) so median is calculated as
Median = The lower boundary of class range + 12/15 x class size= (12/15) x 10)

8. Mode
Age
Frequency
Class Marks
Class width
5-14 7
9.5 10
15-24 15
19.5
25-34 5
29.5
35-44
39.5
45-54
49.5
The modal class is defined to be the class with maximum frequency. The
Mode for grouped data would be the class mark of the modal class.
Mode = 19.5

9. STATISTIC : DESCRIPTIVE MEASURES
Range, Variance, and Standard Deviation
Ungrouped Data
Range
The range for a data set is equal to the maximum value in the data set minus
the minimum value in the data set.
Example :
Range in test score for Johan and Jamal. Range for Johan is 100 – 85 = 15
and the range for Jamal is 90 – 60 = 30. The spread in Johan’s score is as
measure by range twice the spread of Jamal’s score.

10. STATISTIC : DESCRIPTIVE MEASURES
Range, Variance, and Standard Deviation
Ungrouped Data
Variance
s2 =
∑( x – x )2
n-1 _
Variance for sample of size n
δ2 =
∑( x – µ )2 N _
Variance for population of size N

11. STATISTIC : DESCRIPTIVE MEASURES
Range, Variance, and Standard Deviation
Ungrouped Data
Variance: continue…
Example : Times in minutes for 5 students to complete a task were
5, 10, 15, 3 and 7. The mean time is 8 minutes (refer back to mean). See Table
3.0, it illustrates the computation indicated by the formula variance for
sample. _ _
Score
Deviation from mean (x – x )
Squares of deviations (x – x )2 5
5 – 8 = -3
-32 10
10 – 8 = 2 22 15
15 – 8 = 7 72 3
3 – 8 = -5
-52 7
7 – 8 = -1
-12
∑(x – x )=0 (sums of deviations)
∑(x – x )2=88 (sums of squares of deviations) _ _

12. STATISTIC : DESCRIPTIVE MEASURES
Range, Variance, and Standard Deviation
Ungrouped Data
Variance: continue…
If we followed the variance formula in the previous slide, the variance for Table 3.0
is 22 minutes squared. s2 = = = 22
The standard deviation is then calculated as
s = √s2 Sample standard deviation
δ =√δ2 Population standard deviation
The standard deviation is √22 = 4.7 minutes
∑( x – x )2
n-1 _ 88 4

13. STATISTIC : DESCRIPTIVE MEASURES
Coefficient Of Variation
The coefficient variation is equal to the standard deviation divided by the mean.
The result is usually multiplied by 100 to express it as a percent.
The coefficient of variation for a sample is given by
CV = x 100%
The coefficient of variation for population
CV = x 100% s x δ

14. STATISTIC : DESCRIPTIVE MEASURES
Coefficient Of Variation : continue…
Example :
A national sampling of prices for new and used cars found that the mean
price for a new car is $20,000 and the standard deviation is $6,125 and that
the mean price for a used car is $5,485 with a standard deviation equal to
$2,730. In terms of absolute variation, the standard deviation of price for new
cars is more than twice that of used cars. However, in terms of relative variation,
there is more relative variation in the price of used cars than in new cars.
The CV for used cars is and the CV for new cars is
2,730
5,485
X 100 = 49.8%
6,125
20,100
X 100 = 30.5%

15. Exercise : Answer all Question
Q1 . Table below gives the selling prices in tens of thousands of dollars for 20
Homes sold during the past month. Find the mean, mode, and median.
60.5
113.5
79.0
475.5
75.0
70.0
122.5
150.0
100.0
125.5
90.0
175.5
89.0
130.0
111.5
50.0
340.5
525.0
Mean = Mode= Median=

16. Q2 . Find the mean, mode, and median for the grouped data below
Age
Frequency
Class Marks
Class width
20-29 11
30-39 25
40-49 14
50-59 7
60-69 3
Mean = Mode= Median=

17. Q3. Fill in the table below with the details required_ _
Color Car
No involved in accident
Deviation from mean (x – x )
Squares of deviations (x – x )2
Red 10
Blue 5
Yellow
Green 8
Purple 7
∑(x – x )=
∑(x – x )2= _ _
∑( x – x )2
n-1 _
Variance s2 = =