Learning Outcomes: Find and interpret the variance and standard deviation for ungrouped data. (a) Find and interpret the variance and standard deviation for grouped data. (b) (c) Find and interpret the Pearson's Coefficient of Skewness. When coefficient is very close to 0 (negative or positive), the distribution of data is almost symmetrical. At the end of the lesson, students should be able to • Range • Interquartile Range • Variance • Standard deviation Measure of Dispersion Range is NOT a good measure of dispersion because it is influenced by the extreme values and the calculation does not cover all observations. ▪ Variance and standard deviation are more useful and widely used for measure of dispersion. ▪ [Although they are influenced by the extreme values, the calculation cover all the observation] ○Data points are all close to the mean the standard deviation is close to 0. ○ Many data points are far from the mean the standard deviation is far from 0. ○All data values are equal the standard deviation is 0. ▪Standard deviation measures how spreads out the values in a data set are. Remarks: 6.3 Measure of Dispersion C06 -SM025 Page 1
the difference between the observation with the highest value and the observation with the lowest value. For ungrouped data, the range can be determined by: Range the difference between the first quartile and the third quartile. Interquartile Range the average of the squared differences of the data from its mean. For ungrouped data, the variance can be determined by: Variance • For tabulated ungrouped data, variance will be calculated using Remarks: the measure of the amount of variation or dispersion of a set of data. [square root of variance] For ungrouped data, the standard deviation can be determined by: Standard Deviation • For tabulated ungrouped data, standard deviation will be calculated using Remarks: Measure of Dispersion - Ungrouped Data C06 -SM025 Page 2
Data X : Data Y : Calculate the variance and standard deviation for the sets of sample data above. Hence, determine which data is more disperse about its mean. Solution: Data X: Data Y: Since therefore Data Y is more dispersed about its mean than Data X. Example 1 ** C06 -SM025 Page 3
the difference between the upper boundary of the last class and the lower boundary of the first class. For grouped data, the range can be determined by: Range the difference between the first quartile and the third quartile. Interquartile Range the average of the squared differences of the data from its mean. For grouped data, the variance can be determined by: Variance , all the data is very close to mean is far from 0, then many points are far from the mean. , all the data values are equal the measure of the amount of variation or dispersion of a set of data. [square root of variance] For grouped data, the standard deviation can be determined by: Standard Deviation Range is not a good measure of dispersion because it is influenced by the extreme values and the calculation does not cover all observations. Variance and standard deviation are most useful and widely used measure of dispersion. Although they are influenced by the extreme values, the calculations cover all the observations Remarks: Measure of Dispersion - Grouped Data C06 -SM025 Page 4
Find the interquartile range, mean and standard deviation for the data below and interpret the standard deviation. Marks Number of students 9 29 42 26 14 Solution: The first quartile is 34.5 The third quartile is 67.7 The interquartile range is 33.2 The mean is 51.2 The standard deviation is 22.2 are far from the mean. Example 2 [Example 3 in PowerPoint Notes Topic 6.3] C06 -SM025 Page 5
Provides a numerical measure of the skewness of a distribution. The Pearson's coefficient of Skewness be determined by: OR Remarks: If the data have more than 1 mode, then use the formula without involving mode. The strength of skewness: The distribution is symmetrical / no skewed / normal [ ] The distribution is approximately normal. When , the distribution is skewed to the right. When , the distribution is skewed to the left. When , the distribution is symmetrical. Remarks: Pearson's Coefficient of Skewness C06 -SM025 Page 6
2 3 3 4 5 5 6 Determine the Pearson’s coefficient of skewness for the data above. Hence, what can you conclude? Solution: Since , therefore the data are symmetrically distributed. Example 3 C06 -SM025 Page 7
(a) For the above data, find the (i) mean (ii) mode (iii) standard deviation. Class Interval Frequency 3 2 6 4 1 Solution: (i) (ii) (iii) (b) Hence, find and interpret the Pearson's coefficient of skewness. Solution: Since , therefore the distribution is slightly skewed to the left. Example 4 C06 -SM025 Page 8