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NAME:__________________________________________________ Topics in Pre-Calc II – Measures of Central Tendency and Dispersion CLASSWORK DATE:_________ MEASURES OF CENTRAL TENDENCY The mean of a data set is the statistical name for its arithmetic average. ̅ For discrete numerical data, the mode is the most frequently occurring value in the data set. For continuous numerical data, we cannot talk about a mode in this way because no two data values will be exactly equal. Instead we talk about a modal class, which is the class that occurs most frequently. If a set of scores has two modes we say it is bimodal. If there are more than two modes then we do not use them as a measure of the centre. The median is the middle value of an ordered data set. An ordered data set is obtained by listing the data, usually from smallest to largest. If there are n data values, find . The median is the ( ) th data value. For Example: If If , so the median = 7th ordered data value. , so the median = average of the 7th and 8th ordered data values. 1. The mean of the eight numbers listed below is 12. Find the value of x. 5, 9, 11, x, 12, 13, 14, 17 2. The mean and mode of a set of 14 data points below are both 7. Find the values of p and q. 6, 8, 7, 7, 5, 7, 6, 8, 6, 9, 6, 7, p, q MEASURES OF DISPERSION RANGE – the difference between the highest and lowest number in a set of data QUARTILES – separates the data into 4 equal parts *1st Quartile(Lower Quartile)- 25% of the data values are less than or equal to the lower quartile *2nd Quartile- (Median) - 50% of the data values are less than or equal to the median *3rd Quartile- (Upper Quartile) - 75% of the data values are less than or equal to the upper quartile 58, 60, 65, 70, 72, 75, 76, 80, 80, 81, 83, 84, 86, 87, 87, 90 INTERQUARTILE RANGE – the difference between the first and third quartile values STANDARD DEVIATION – measure of how spread out the numbers are. In other words, it represents the “distance” from the mean for a certain set of data. The standard deviation can be used to show consistency of your data. - Big standard deviation - If the data is spread out, the standard deviation will be large (less consistency) Small/low standard deviation - If the data is close together, the standard deviation will be small (more consistency) CALCULATING MEASURES OF CENTRAL TENDENCY and DISPERSION IN THE CALCULATOR NO FREQUENCY TABLE FREQUENCY TABLE 72, 89, 41, 89, 73, 72, 91 1. Enter the data into the calculator 2. Calculate the data sets of the data entered into the calculator 1. Enter the data into the calculator 3. Identify the appropriate data sets from the calculator 2. Calculate the data sets of the data entered into the calculator 3. Identify the appropriate data sets from the calculator 3. For the given data, find all the following data sets: (a) (b) (c) (d) (e) (f) (g) (h) State the mode State the median State the mean State the population standard deviation State the range State the lower quartile State the upper quartile State the interquartile range 4. The following table shows the age distribution of teachers who smoke at Laughlin High School. Ages Number of smokers 20 ≤ x < 30 5 30 ≤ x < 40 4 40 ≤ x < 50 3 50 ≤ x < 60 2 60 ≤ x < 70 3 (a) State the modal interval of the given data. (b) Find the mean of the given data. (c) State the median interval of the given data. (d) State the population standard deviation. BOX AND WHISKER PLOTS How do you enter this data into your calculator if they give you an interval for the x-values? 5. The box plot below summarises the points scored by a netball team. Find the following data sets: (a) (b) (c) (d) (e) (f) (g) The median The maximum value The minimum value The upper quartile The lower quartile The range The interquartile range 6. The weight in kilograms of 12 students in a class are as follows. 63 76 99 65 63 51 52 95 63 71 65 83 Draw a box-and-whisker plot of the data using the axis below. 50 60 70 80 90 100