1. Chapter 3 – Data Description Section 3.1 – Measures of Central Tendency

2. Measures of Central Tendency “’Average’ when you stop to think about it is a funny concept. Although it describes all of us it describes none of us…While none of us wants to be the average American, we all want to know about him or her.” The average American man is 5’9” tall. The average American woman is 5’3.6”. The average American is sick in bed 7 days a year missing five days of work. On the average day, 24 million people receive animal bites By his or her 70 th birthday, the average American will have eaten 14 steers, 1050 chickens, 2.5 lambs and 25.2 hogs.

3. Measures of Central Tendency Average does not mean exactly what you think it means… An average could indicate several different things Mean Mean Median Median Mode Mode Midrange Midrange Also, an ‘”average” could be describe a sample or a population

4. Measures of Central Tendency Statistic – a characteristic or measure obtained by using data values from a SAMPLE. a characteristic or measure obtained by using data values from a SAMPLE. We generally use ROMAN letters to represent statistics (A, B, C, D …) We generally use ROMAN letters to represent statistics (A, B, C, D …) Parameter – a characteristic or measure obtained by using data values from a POPULATION. a characteristic or measure obtained by using data values from a POPULATION. We generally use GREEK letters to represent parameters (σ, β, α, µ …) We generally use GREEK letters to represent parameters (σ, β, α, µ …)

5. Measures of Central Tendency Mean – the sum of the values, divided by the total number of values. the sum of the values, divided by the total number of values. this is usually what you are talking about when you say average this is usually what you are talking about when you say average x-bar is used to represent the sample mean. µ is used to represent the population mean.

6. Measures of Central Tendency The Median (MD) arrange observations from smallest to largest. median is either the middle number or the mean of the middle two numbers.

7. Measures of Central Tendency The Mode the data value that occurs most frequently the data value that occurs most frequently Unimodal- data set has only one value with the greatest frequency. Unimodal- data set has only one value with the greatest frequency. Bimodal- data set has 2 values with the greatest frequency. Bimodal- data set has 2 values with the greatest frequency. Multimodal- data set has more than 2 values with the greatest frequency. Multimodal- data set has more than 2 values with the greatest frequency. No mode- no data value occurs more than once No mode- no data value occurs more than once

8. Measure of Central Tendency Modal Class- class with the largest frequency Pg. 112 example 3-12

9. Measures of Central Tendency The Midrange the mid point of a data set the mid point of a data set MR= (min + max) / 2 MR= (min + max) / 2

10. Measures of Central Tendency Rounding Rules – In general, round to one place after the last place given in the data. In general, round to one place after the last place given in the data. ex. 3.45, 5.21, 6.89, 4.22 ex. 3.45, 5.21, 6.89, 4.22 round to three decimal places. round to three decimal places.

11. Measures of Central Tendency The Median and Mode are resistant, meaning unusually large or small values do not affect it. The Mean and Midrange are not. The one huge house in the neighborhood allows the mean home value to skyrocket.

12. Measures of Central Tendency GROUPED DATA When data is grouped in a distribution or in a graph things are slightly different. First, make a distribution table with these column headings Class Class Frequency Frequency Midpoint Midpoint Then add one more column frequency times midpoint frequency times midpoint

13. Measures of Central Tendency GROUPED DATA Look at the procedure table on pg. 108 Finding the mean for grouped data

14. Measures of Central Tendency GROUPED DATA Turn to page 107 and look at example 3-3. Remember: Midpoint = Upper limit + lower limit / 2 Remember: Midpoint = Upper limit + lower limit / 2 You will add your f●x m altogether and then divide by the total frequency (n). You will add your f●x m altogether and then divide by the total frequency (n).

15. Measures of Central Tendency GROUPED DATA Weighted Mean found by multiplying each value by its corresponding “weight” and dividing the sum of the products by the sum of the weights. found by multiplying each value by its corresponding “weight” and dividing the sum of the products by the sum of the weights. For example Grade Point Averages in College Grade Point Averages in College A = 4 points, B = 3 points, C = 2 points, D = 1 point. A = 4 points, B = 3 points, C = 2 points, D = 1 point. Each class has a different weight… Each class has a different weight…

16. Example 3-17 pg. 115 ClassCredits (w)Grade (x)wX English Composition 13A (4 points) Introduction to Psychology3C (2 points) Biology 14B (3 points) Physical Education2D(1 point) X = ΣwX Σw =

17. Distribution Shapes (pics pg. 59) Bell-Shaped- a single peak and tapers off at either end Uniform- flat or rectangular J-Shaped- few data values on left side, increases from left to right Reverse J-shaped- few data values on right side, decreases from left to right Bimodal- has 2 peaks of the same height U-shaped- shaped like a U

18. Distribution Shapes (pg. 117 for pics) Positively Skewed or Right-Skewed Majority of data values fall on the left of the mean and cluster at the lower end Majority of data values fall on the left of the mean and cluster at the lower end Tail is to the right Tail is to the right Symmetric Distribution Data values evenly distributed on both sides of the mean Data values evenly distributed on both sides of the mean Negatively Skewed or Left-Skewed Data values fall on right of the mean and cluster at the upper end. Data values fall on right of the mean and cluster at the upper end. Tail is to the left Tail is to the left

19. Practice! Pg. 118 2, 3, 7, 10, 12, 13 2, 3, 7, 10, 12, 13