8.2 Measures of Central Tendency and Variation

 

Grouped DATA – Activity 4 – Page 174 in Activity Book

 

READ TEXT!!  Nice Example of 3 teachers comparing for "best class" page 525

 

My newspaper statistics

 

Desire to represent with fewer numbers

 

Measures of "CENTER" (Average, typical – mean, median, and mode)

 

Measure of "SPREAD"  (range, variance, standard deviation, interquartile range)

 

Measures of CENTRAL TENDENCY

 

Activity 5 in Activity book, pg 177

 

MEAN  (average or arithmetic mean)     

Sum of numbers divided by the number of items

 

The Arithmetic Mean of the numbers , denoted  and read ”x bar,”

         is given by  

 

          Balance point  - pictures pages 526 & 527

 

 

The tram at a ski area has a capacity of 50 people with a load limit of 7500 lb.  What is the mean weight of the passengers if the tram is loaded to capacity?

 

 

 

 

If the mean weight of seven tackles on a team is 230 lb and the mean weight of the four-backfield members is 190 lb, what is the mean weight of the 11-person team?

 

 

 

 

 

MEDIAN    the value exactly in the middle of and ORDERED set of  data

 

Arrange in order!!!  Then middle number

         

          If an odd number of values, then the median is the “middle number”

If an even number of values, then average the middle 2 values

 

 

MODE       Most frequently occurring value, if there is one

 

          Sometimes, bimodal

          Often the mode is not a good measure of the center

 

Examples

 

a)    1, 4, 11, 19, 2, 3, 7, 2, 16

 

 

 

 

 

b)    4, 114, 7, 114, 10, 19, 11, 17, 12, 15, 14

 

 

 

 

 

 

c)    16, 2, 2, 14, 2, 3, 13, 4, 7, 10

 

 

 

 

 

 

If mean for group of 24 people is $8000, what is the mean if one additional person whose income is $58,000 enters the group?

 

 

 

 

 

Which Measure of Center Most appropriate?  (Good examples pg 530)

          Mean - Distortion of "ouliers" – salaries

                     effected by extreme values

                     most commonly used as average

 

          Median - not effected by extremes

 

          Mode - only one score effects

 

Suppose you own a hat shop and decide to order hats in only one size for the coming season.  To decide which size to order, you look at last year’s sales figures, which are itemized according to size.  Should you find the mean, median, or mode for the data?  Why?

 

 

 

 

Measures of SPREAD or DISPERSION   -  How data is spread out

 

RANGE  =  High - Low  -  only involves 2 scores

 

 

BOX PLOTS or Box and Whisker Plots

 

          Five Number Summary

 

          Arrange in order

 

a)  find lowest, highest and median

 

b)  Split in 2 at median into 2 halves and find median of each

 

                             Upper & Lower Quartile

 

          InterQuartile Range  (IQR) – Range of middle half of data

 

                   Upper Quartile - Lower Quartile

 

          Draw the IQR as BOX and whiskers to max and min

 

          Outliers     Far away from the rest of data  -   indicate with a *

                Any value that is more than 1.5 times the interquartile range above the

                    upper quartile or below Lower Quartile

Construct a box-and-whisker plot for the following set of test scores.  Indicate outliers, if any, with asterisks.

          20  95  40  70  90  70  80  80  90  95

 

 

 

 

 

 

 

 

 

 

Compare Sets of Data  - Use Stem & Leaf   but sometimes Box plots better

                   Example pgs 535 & 536 of male vs female salary over time

 

 

Mean Absolute Deviation (MAD) – is average distance from the mean

          Find differences from the mean    

          Make all positive, i.e., take their “absolute value”

          Average the positive differences

 

 

 

What does it mean if the mean absolute deviation is zero?

 

 

 


VARIANCE and STANDARD DEVIATION

 

 

          Steps in box

1.    Find the Mean

2.    (x - mean) for each value

3.    square each value in #2

4.    sum values in #3

5.    Divide by n to get the variance v

6.    Square root of variance is standard deviation

Interpret Standard Deviation

    Large Standard Deviation - means more spread out

    Small Standard Deviation – means data grouped tightly together

 

Examples: 

Find Range, MAD, Variance and Standard Deviation for each set of values

3       4  7  12  13  15

 

 

 

 

 

 

 

 

 

 

24   29  35  39  43  48  55  56  71  79  85  96

 

 

 

 

 

 

 

 

 

Standard Deviation from a grouped frequency table

 

5, 5, 5, 7, 7, 7, 7, 7, 10, 10 ,10, 10, 10, 10, 10, 12, 12, 12, 12, 32

       becomes table below

 

   X

f

 5

3

 

 

 

 

 7

5

 

 

 

 

10

7

 

 

 

 

12

4

 

 

 

 

32

1

 

 

 

 

 

 

Normal Distributions  -  Normal Curve –

 

*Our text uses 99.8 for  “within three standard deviations”

 

Mean, median and mode all have the same value.

 

Sugar Plops boxes say they hold 16 oz.  To make sure they do, the manufacturer fills the box to a mean weight of 16.1 oz, with a standard deviation of 0.05 oz.  If the weights have a normal curve, what percentage of the boxes actually contain 16 oz or more?

 

 

According to psychologists, IQs are normally distributed with a mean of 100 and a standard deviation of 15.   What percentage of the population have IQs lower than 85?

 

 

On a certain exam, the mean is 72 and the standard deviation is 9.  If a grade of A is given to any student who scores at least two standard deviations above the mean, what is the lowest score that a person could receive and still get an A?

 

 

A tire company tested a particular model of tire and found the tires to be normally distributed with respect to wear.  The mean was 28,000 mi. and the standard deviation was 2500 miles.  If 2000 tires are tested, about how many are likely to wear out before 23,000 mi.?

 

 

Assignment:  8.2  pg. 545 – 2, 3, 5, 6, 7, 10, 13a, 14, 16, 17, 18a,c, 22, 23 a,c, 28, 31, 32, 34, 36, 37, 39, 40, 43, 45 47, 62, 63, 65a, b, Both NAEP