بسم الله الرحمن الرحيم 1

Probability and Significance

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

Features of normal distribution curve

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

Normal Distribution Curve - 2 SD -1SD Mo +1SD +2 SD

Unimodal Curve

Bimodal Curve

Normal Distribution Curve Negative tail Positive tail

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

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

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

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

Symmetric

Asymmetric

Normal Curve Empirical Rule

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

Confidence Limit

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

Confidence Limit The confidence limit Negative tail Positive tail

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

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

Region of Rejection  

Probability and Significance

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

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

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

Probability & Significance If P > 0.01 ( highly significantly different from the curve → Highly significant ) If P > 0.001 ( Very highly significantly different from the curve → Very highly Significant )

Revision Activities

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

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

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

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

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

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

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? 2 - How many standard deviations , a student of 1.85 m is from the mean ?

The End Thanks