1. بسم الله الرحمن الرحيم 1

2. Probability and Significance

3. Topics to be covered Features of normal distribution curve
Normal curve empirical rule
Confidence limit
Probability and significance
Revision activities

4. Features of normal distribution
curve

5. Normal Distribution Curve
Is a theoretical , bell shaped unimodal curve that extends to + / - infinity & not touch base line
Completely Described by Two Parameters , mean & standard deviation

8. Normal Distribution Curve
- 2 SD SD Mo SD SD

9. Unimodal Curve

10. Bimodal Curve

11. Normal Distribution Curve
Negative tail
Positive tail

12. Normal Distribution Curve
- The shape of the distribution curve depends on the scale of plotting of the frequency
- The frequency can be plotted using histogram & frequency polygon
- If the 2 tails of the curve are equal , the curve is symmetric but if one tail is longer than the other , the curve is skewed or asymmetric

13. Normal Distribution Curve
- If the mean & median values are equal or nearly equal , data are symmetrically distributed
- If the mean value is less than the median , data are positively skewed
- If the mean value is more than the median , data are negatively skewed

14. Ungrouped data Mean = Ʃ x / n Grouped data Mean = Ʃ f X / n
Mean Equation
Ungrouped data Mean = Ʃ x / n Grouped data Mean = Ʃ f X / n
Where f refers to the frequency and x refers to the observations and n is the number of observations

15. Median : is the number that bisects the observations into equal values
Median Equation
Median : is the number that bisects the observations into equal values
To obtain the median , arrange the values
by order from the highest to lowest or from the lowest to highest then the order of the median value will be : ( n + 1 ) / 2 If the data are odd number ( n / 2 ) & ( n/ 2 ) + 1 If the data are even number

16. Symmetric

17. Asymmetric

18. Normal Curve Empirical Rule

19. Normal Distribution Curve
The empirical rule states that for a normal distribution
68% of the data will fall within 1 standard deviation of the mean
95% of the data will fall within 2 standard deviations of the mean
(99.7%) of the data will fall within 3 standard deviations of the mean

22. Confidence Limit

23. Confidence Limit
X ± 1 SD = 68% of observations
X ± 2 SD = 95 % of observation and this is the confidence limit

24. Confidence Limit
The confidence limit
Negative tail
Positive tail

26. If you go back to the previous slide you will find that :
Confidence Interval
If you go back to the previous slide you will find that :
X = 1.4
Upper limit cut off point (X + 2SD ) = 1.7
Lower limit cut off point (X - 2SD ) = 1.1
The area highlighted red is the area between upper & lower limit cut off points and it is the confidence interval

27. Confidence Limit & Interval
You are confident that any data within the range of mean ± 2 SD are within the curve
Example : the mean age is 50 years and the SD is 10 , so the age range 30 – 70 years is confidently lying under the curve

28. Region of Rejection  

29. Probability and Significance

30. Probability & Significance
Probability ( P ) Is the probability of being under the curve
If P is less than 5% , this means that the observation is outside the confidence limit so it is significantly different from the curve

31. Probability & Significance
Probability ( P ) Is the probability of being under the curve
If P is more than 5% , this means that the observation is inside the confidence limit so it is insignificantly different from the curve

32. Probability & Significance
If P > 0.05 ( not significantly different from the curve → Insignificant )
If P > 0.05 ( significantly different from the curve → Significant )

33. Probability & Significance
If P > 0.01 ( highly significantly different from the curve → Highly significant )
If P > ( Very highly significantly different from the curve → Very highly Significant )

34. Revision Activities

35. Activity 1
The following are three groups of height data with the universe mean = 160 cm , Please construct histogram for each group and comment on the distribution of these data whether symmetrically or asymmetrically distributed

36. Group 1
Frequency
Height 2
155 8
160 12
175 20
185 27
190 32
195 17
200
205

37. Group 2
Frequency
Height 28
120 37
125 40
130 33
135 22
140 13
150 7
160 3
165

38. Group 3
Frequency
Height 7
130 18
140 34
150 45
160 29
170 12
180 5
190

39. Activity 2
Using the mean and median principle , please identify whether the following groups of data are symmetrically or asymmetrically distributed

40. Group 3
Group 2
Group 1
130
120
155
140
125
160
150
175
135
185
170
190
180
195
200

41. Activity 3 95% of students at school are between 1.1m and 1.7m tall.
1 - Assuming this data is normally distributed can you calculate the mean and standard deviation? How many standard deviations , a student of m is from the mean ?

42. The End Thanks