Chapter 3:Data Description

Learning Target

IWBAT summarize data, using measures of central tendency, such as the mean, median, mode, and midrange.

Vocabulary

Statistic› A characteristic or

measure obtained by using the data values from a sample

Parameter› A characteristic or

measure obtained by using all the data values from a specific population

The Mean

Mean is also known as an arithmetic average. It is found by adding the values of the data and dividing by the total number of values.

Sample Mean and Population Mean

Sample Mean Population Mean

Rounding Rules

General Rule:› Wait until the end to round. All calculations

in between should not be rounded. Rule for Mean:

› The mean should be rounded to one more decimal place than the raw data.

Grouped Frequency Means Using the frequency

distribution, find the mean. A

ClassB

Frequency (f)

CMidpoint (Xm)

D(f)(Xm)

5.5-10.5 1 8 8

10.5-15.5

2 13 26

15.5-20.5

3 18 54

20.5-25.5

5 23 115

25.5-30.5

4 28 112

30.5-35.5

3 33 99

35.5-40.5

2 38 76

n = 20 Sum=490

Step 1: Create table

Step 2: Find the midpoints of each class.

Step 3: For each class, multiply the frequency by the midpoint.

Step 4: Find the sum of column D.

Step 5: divide the sum by n to get the mean.

The Median

The midpoint of the data Symbol for median is MD To find median

› Step 1: arrange data in order› Step2 : select the middle point

› If there are 2 middle numbers, add the numbers and divide by 2.

The Mode

The number that occurs most often Types of modes

› No mode› Unimodal – one mode› Bimodal – 2 modes› Multimodal – more than 2 modes

› The mode for grouped data is the modal class. The modal class is the class with the largest frequency.

The Midrange

The midrange is the sum of the largest value and the smallest value, divided by 2.

The symbol for midrange is MR.

The Weighted Mean

To find the weighted mean, multiply each value by its corresponding weight and divide the sum of the products by the sum of the weights.

ExampleCourse Credits

(w)Grade (X)

English 3 A (4 points)

Intro to Psychology

3 C (2 points)

Biology I 4 B (3 points)

Physical Education 2 D (1 point)

The grade point average is 2.7.

Handouts

Section 3-2

Measures of Variation

Learning Target

IWBAT describe data, using measures of variation, such as the range, variance, and standard deviation.

Range

The range is the highest number minus the lowest number. It is represented by R.

One extremely high or one extremely low number can affect the range.

Population Variance and Standard Deviation

The variance is the average of the squares of the distance each value is from the mean.

Symbol is 2

Formula is › 2 = (-)2/› is lowercase sigma› is the individual

value› is the mean› is the number of data

values

The standard deviation is the square root of the variance.

Symbol is . Formula is

› = 2 = (-)2/

Steps to Find 2 and

Step 1: Find the mean for the data. Step 2: Subtract the mean from each

data value. Step 3: Square each result. Step 4: Find the sum of the squares. Step 5: divide the sum by N to get the

variance. Step 6: Take the square root of the

variance to find the standard deviation.

Example:Find the variance and standard deviation of

the following data: 10, 60, 50, 30, 40, 20.Step 1: Mean is (10+60+50+30+40+20)/6

210/6 = 35Step 2: Subtract mean from each data value

10 - 35 = -25 60 – 35 = 25 50 – 35 = 15

30 – 35 = -5 40 – 35 = 5 20 – 35 = -15

Step 3: Square each result.(-25)2 = 625 (25)2 = 625 (15)2 =

225(-5)2 =25 (5)2 = 25 (-15)2

=225

Example (cont’)

Step 4: Find the sum of the squares. 625+625+225+225+25+25 = 1750

Step 5: Divide the sum by N to get the variance.Variance = 1750/6 = 291.7

Step 6: Take the square root to get the standard deviation. 291.7 = 17.1

Variance and Standard deviation will never be negative.

Practice Problem

Find the Variance and Standard Deviation of the following data.

35, 45, 30, 35, 40, 25

Answer

Variance is 250/6 = 41.7 Standard deviation = 6.5

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When the means are equal, we need to do more tests to analyze the data. Therefore, we can determine the variability of the data by finding the variance and the standard deviation. Variability is also known as the spread of the data. The larger the variance and the standard deviation the more variable the data is or spread out the data is.

For example, in the manufacture of fittings, such as nuts and bolts, the variation in diameter must be small, or parts will not fit together.

The standard deviation also tells you how far the data is away from the mean.

Variance and Standard Deviation of Grouped Data

Process is similar to finding the mean for grouped data.

Step 1: Make a table Step 2: Multiply the midpoint by the frequency. Step 3: Multiply the frequency by the square of the

midpoint. Step 4: Find the sum of the frequency, step 2 (m),

and step 3 (m2).

Step 5: Find the variance by using this formulas2 = n( m

2 )-( m)2

n(n-1) Step 6: Take the square root of step 5 to find the

standard deviation.

ExampleClass Frequenc

yMidpoint

(Xm)m m

2

5.5 – 10.5 1 8 8 64

10.5 – 15.5 2 13 26 338

15.5 – 20.5 3 18 54 972

20.5 – 25.5 5 23 115 2645

25.5 – 30.5 4 28 112 3136

30.5 – 35.5 3 33 99 3267

35.5 – 40.5 2 38 76 2888

n = 20 m = 490 m2 = 13310

s2 = n( m2

)-( m)2 s = 68.7 = 8.3n(n-1)

s2 = 20(13310) – (490)2

20(20-1)s2 = 266200 – 240100

20(19)s2 = 26100 380s2 = 68.7

Practice Problem

Comparing Standard Deviations when units are different

If the units of two sets of data are different we can use the coefficient of variation to compare the standard deviations.

The coefficient of variation is the standard deviation divided by the mean and is expressed as a percent.

Symbol for the coefficient of variation is CVar.

Formula is CVar = s/X times 100%

Example

The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variations of the two.

Sales = 5/87 times 100% = 5.7% Commissions = 773/5225 times 100% =

14.8% The commissions are more variable than the

sales.

Practice Problem

The mean for the number of pages of a sample of women’s fitness magazines is 132, with a variance of 23; the mean for the number of advertisements of a sample of women’s fitness magazines is 182, with a variance of 62. Compare the variations.

Answer

The coefficients of variation are› Pages = 23/132 times 100% = 3.6%› Advertisements = 62/182 times 100% =

4.3%› The number of advertisements is more

variable than the number of pages.

Worksheets

Pg. 137›1, 2, 16, 18 – 25, 27 – 31

Chebyshev’s Theorem

The proportion or percent of values from a data set that will fall within k standard deviations of the mean will be at least 1-1/k2, where k is a number greater than 1. (k is also the number of standard deviations)