Study of Measures of Dispersion and Position

DATA

QUALITATIVE QUANTITATIVE

DISCRETECONTINUOUS

Cannot be given a numerical valueExamples:Gender, nationality, television show

preference

Can be given and analyzed as numerical values

Examples: test scores, weights of objects, hours studied

A type of data is discrete if there are only a finite number of values possible or if there is a space on the number line between each 2 possible values.

Example: A 5 question quiz is given in a Math class. The number of correct answers on a student's quiz is an example of discrete data. The number of correct answers would have to be one of the following : 0, 1, 2, 3, 4, or 5. There are not an infinite number of values, therefore this data is discrete. Also, if we were to draw a number line and place each possible value on it, we would see a space between each pair of values.

Example. In order to obtain a driver’s license a person must pass a written exam. How many times it would take a person to pass this test is also an example of discrete data. A person could take it once, or twice, or 3 times, or… . So, the possible values are 1, 2, 3, … . There are infinitely many possible values, but if we were to put them on a number line, we would see a space between each pair of values.

Continuous data makes up the rest of numerical data. This is a type of data that is usually associated with some sort of physical measurement.

Ex. The height of individuals is an example of continuous data. Is it possible for a person to be 5'2" tall? Sure. How about 5'2.5" ? How about 5'2.525“? The possibilities depends upon the accuracy of our measuring device.

One general way to tell if data is continuous is to ask yourself if it is possible for the data to take on values that are fractions or decimals. If your answer is yes, this is usually continuous data.

Ex. The length of time a battery works is an example of continuous data. Could it work 200 hours? How about

200.7? 200.7354?

Many continuous variables have data distributions that are bell-shaped

Ex: heights of adults, body temperature of animals, cholesterol levels of adults

Ex: data collected a) height of 100 women

b) Increase the sample size and decrease the intervals

c) Continue to increase and decrease

d) Normal distribution for the entire population

A normal distribution

is symmetric about the

mean.

What are some other univariate data (data with one variable) that can be modeled using a normal distribution?

NEGATIVELY SKEWED POSITIVELY SKEWED

-when the majority of the values fall to the right of the mean -the mean is to the left of the median, and the mean and the median are to the left of the mode

-When the majority of the data values fall to the left of the mean-The mean falls to the right of the median, and both the mean and the median fall to the right of the mode

Visually, what indicates the direction of the skewness?Skewness is the degree of departure from symmetry of a distribution. A positively skewed distribution has a "tail" which is pulled in the positive direction. A negatively skewed distribution has a "tail" which is pulled in the negative direction.

Kurtosis is the degree of peakedness of a distribution. A normal distribution is a mesokurtic distribution. A pure leptokurtic distribution has a higher peak than the normal distribution and has heavier tails. A pure platykurtic distribution has a lower peak than a normal distribution and lighter tails.

Kurtosis is the degree of peakedness of a distribution. A normal distribution is a mesokurtic distribution. A pure leptokurtic distribution has a higher peak than the normal distribution and has heavier tails. A pure platykurtic distribution has a lower peak than a normal distribution and lighter tails.

Measures of variation show the spread of the data. Quartiles and the interquartile range describe the spread in the middle half of the data. Mean absolute deviation, variance and standard deviation describe the spread of the data around the mean. Two sets of data may have the same range and mean, but the spread of the data can be very different.

Data can be represented by measures of central tendency and measures of variation, such as range, quartiles and the interquartile range.

Summation Notation

µ = arithmetic mean of a population = arithmetic mean of a sample

= Variance = standard deviation

Formula:

Definition: The average of the absolute values of the differences between the mean and each value in the data set.

(Determines the average distance from an occurrence to the mean)

Step 1: Find the meanStep 2: Find the sum of the absolute values of

the differences between each value in the set of data and the mean.

Step 3: Divide by the number of values in the set.

1

n

ii

x

n

The top 10 finishing times (in seconds) for runners in two men’s races are as given. The times in a 100 meter dash are in set A and the times in a 200 meter dash are in set B.

Set A Set B

10.62 21.37

10.94 21.40

10.94 11.23

10.92 22.23

11.05 22.34

11.13 22.34

11.15 22.36

11.28 22.60

11.29 22.66

11.32 22.73

Compare the spread of data for the two sets using the range and the mean absolute deviation.

1

n

ii

x

n

MAD =

Set A

10.62

10.94

10.94

10.92

11.05

11.13

11.15

11.28

11.29

11.32

1

n

ii

x

n

Mean = ______

Set B

21.37

21.40

11.23

22.23

22.34

22.34

22.36

22.60

22.66

22.73

1

n

ii

x

n

Mean = ______

For any bell-shaped curve, approximately: 68% of the values fall within one standard deviation of the mean

in either direction 95% of the values fall within 2 standard deviations of the mean in

either direction 99.7% of the values fall within 3 standard deviations of the mean

in either direction

to determine the spread of data. If the variance or standard deviation is large, the data are more dispersed. This information is useful in comparing two (or more) data sets to determine which is more (most) variable).

to determine the consistency of a variable. For example, in the manufacture of fittings, such as nuts, bolts, the variation in the diameters must be small, or the parts will not fit together.

to determine the number of data values that fall within a specified interval in a distribution.

The variance and standard deviation are used quite often in inferential statistics.

-Is the square root of the variance-“the average distance values fall from the mean-It measures the variability, by summarizing how far individual data values are from the mean.

SET

Numbers Mean Standard Deviation

1 100,100,100,100,100 100 0

2 90,90,100,110,110 100 10

Given the definition, what is the formula for standard deviation:

2

1

n

ii

x

n

Standard Deviation of a Population Data Set

Example: Renee surveyed his classmates to find out how many hours of exercise each student did per week. Find the standard deviation of the data set to the nearest tenth:

3, 10, 11, 10, 9, 11, 12, 8, 11, 8, 7, 12, 11, 11, 5

Example: Josie wants to see if she is charging enough for a babysitting job. She charges $7.50 per hour. She surveyed her friends to see what they are charging per hour. The results are: $8, $8.50, $9, $7.50, $10, $8.25 and $8.75. Determine the mean absolute deviation and use the result to determine if Josie should change her babysitting rate. Explain your reasoning.

Example: Mr. Martin keeps track of the amount of text messages his son sends per month. He feels that his son should spend

less time texting and more time studying his algebra. As an incentive to do more studying, Mr. Martin has agreed to purchase a new computer for his son if he texts more than 1 standard deviation away from the mean. Use the current data set to determine to determine the amount of texts Mr. Martin’s son may send.

Month Messages

October 985

November 1005

December 1100

January 950

February 1200

March 1010

Can measure in terms of actual data distance units from the mean.

Measure in terms of standard deviation units from the mean.

z-score standard measureix

Why do that?So we can compare elements from two So we can compare elements from two

different data sets relative to the different data sets relative to the position within their own data set.position within their own data set.

Consider this problem… Amy scored a 31 on the mathematics

portion of her 2009 ACT® (µ=21 σ=5.3). Stephanie scored a 720 on the

mathematics portion of her 2009 SAT® (µ=515 σ=116.0). Whose achievement was higher on the

mathematics portion of their national achievement test?

Amy

Stephanie

1.89 vs. 1.77 What Does This Mean?