2.3.2 Measures of Spread

Standard Deviation, σ

Measures of Spread

• Range– Measures spread of ordered data between first and

last points

• IQR– Measures spread of ordered data – Middle 50% of data– Compared to range– How tightly clustered data is to the median– Use box plots to visualize

Standard Deviation

• IQR is effective, but awkward to calculate

• Has limited effectiveness

• Recall deviation– Distance of particular piece of data from the mean

• Variance

2

2 ix x

n

Standard Deviation• Much more useful measure than IQR

– Takes all data into account

• Square of variance

• Tells how tightly clustered data is to mean

Greek sigma (lower case)

2

ix x

n

Standard Deviation

• averages square of distance that each datum is from the mean

• If most data clustered around mean– Little dispersion– Small

• If data widely scattered– Lots of dispersion– Large

Note:

• The formula we will be using in this course is for the standard deviation for population data

• You might also see standard deviation for sample data

2

1ix x

sn

• “Small/large σ” is relative to mean and range

Example 1

Ted and Aasma are filling french fry containers. They record how many fries they scoop into each container.

1st 2nd 3rd 4th 5th 6th

Ted 34 41 40 38 38 45

Aasma 51 28 36 44 41 46

Which worker is more consistent?

Example 1 (cont.)

• Ted– Calculate mean first

ixx

n

34 41 40 38 38 45

6

39.3

Example 1 (cont.)

• Ted (continued)– Now calculate standard deviation

2

ix x

n

3.35

2 2 2 2 2 2(34 39.3) (41 39.3) (40 39.3) (38 39.3) (38 39.3) (45 39.3)

6

Example 1 (cont.)• Wow! That’s tedious!

• Use the computer to find the other.

• We get Aasma’s standard deviation is approximately 7.39.

• Who is more consistent?

• Ted– His standard deviation is smaller– His data are more closely clustered about the mean– He is a more consistent French-fry-box-filler

Working with Grouped Data

Ms. McPhee really likes Smarties, but each box does not necessarily contain the same number of candies. The following table represents a sample of 31 boxes showing the number of Smarties per box.

Number of Smarties

28 29 30 31 32 33

Frequency 2 5 10 9 4 1

Calculate the standard deviation for this sample.

Grouped Data• Calculate mean first:

• The mean is approximately 30.4 Smarties

• Next, calculate the standard deviation:

2

if x x

n

1.2

Pg. 170 #7

• Open “2.3.2 Pulse rates.ftm”