Standard Deviation

Warm-up

Do girls study more than boys? We asked the students in three AP Statistics classes how many minutes they studied on a typical weeknight. Here are the responses of random samples of 30 girls and 30 boys from the classes:

WARM-UP

1. Construct a back-to-back boxplot for this

data.2. Compute for the numerical summary of

this data. 3. Find the outliers for both male and female

respondents.

4. Describe the distribution of each data

set by looking at your boxplots. What will be your conclusion about the male/female study

habits base on your statistical data.

Standard Deviation

The standard deviation measures spread by looking at how far the observations are from their mean

Example

A population consists of four observations: {1, 3, 5, 7}. What is the variance? What is the standard deviation?

x =1+3+5+74

= 4

s2=(1-4)2 + (3-4)2 +(5-4)2 +(7-4)2

(4-1)

s2 =(-3) 2 + (-1) 2 +(1) 2 +(3) 2

(3)

20

3= 6.67

s2 9+1+1+9

3=

s=√ 6.67

s= 2.58

Variance

Standard

Deviation

Properties of Standard Deviation

s measures spread about the mean and should be used only when the mean is chosen as the measure of center.

• s = 0 only when there is no spread/variability. This happens only when all observations have the same value. Otherwise, s > 0. As the observations become more spread out about their mean, s gets larger.

s, like the mean x, is NOT resistant. A few outliers can make s very large.

Choosing Measures of Center and Spread

5-number summary VS x and sThe 5-number summary is usually

better than the mean and standard deviation for describing a skewed distribution with strong outliers.

5-number summary x and s

SThe distribution of the C-section performed by the MALE doctors is more skewed to the right while for the female doctors has approximately symmetrical

distribution

O

C

S

Two male physicians performed an unusually high number of cesarean sections, 85 and 86. I used the 1.5xIQR rule to verify these outliersThe mean and median numbers of cesarean sections performed are higher for the male doctors. More than half of the female doctors in the study performed fewer than 20 cesarean sections in a year. Both the standard deviation and the IQR for the male doctors are much larger than the corresponding statistics for the female doctors. So there is much greater variability in the number of cesarean sections performed by male doctors

INTERPRETING NUMERICAL DATA

Statistical toolsDescribing SOCS and skewness

Constructing dotplots and stemplots

Describing the SOCS of your distribution

Creating histogram

Generating histogram using graphing calculator

Organizing data by FDT

Constructing ogive and finding percentile using ogive.