1. Statistics 200 Lecture #4 Thursday, September 1, 2016
Textbook: Sections 2.6 through 2.7
Objectives (relating to quantitative variables):
• Use side-by-side boxplots to explore relationship between quantitative and categorical variable
• Understand intuition for two basic measures of spread:
– IQR (interquartile range)
– Standard deviation
• Learn and apply empirical rule for bell-shaped (normal) distributions

2. Yesterday: Descriptive Methods for Quantitative Data
Graphs:
histogram,
boxplot
numbers: statistics
measures
of center:
mean, median

3. Sex Side by side boxplots:
Compare the distributions of two or more groups
Include an
explanatory
variable:________
Sex

4. Side by side boxplots:
Compare the distributions of two or more groups

5. Example 3: Comparing Two or More Boxplots - What to look for
Part of Graph
Comparisons
Conclusions
within box
compare ______
(center)
smallest:
largest:
compare spread
(IQR)
compare shapes
slightly right skewed:
left skewed:
outside box
compare number of outliers
least:
most:
Median
Female
Male
Female
Male M
Female
Male
Female

6. Measures of Spread (Variation)
Standard Deviation
Interquartile Range (IQR)
Resistant
Sensitive
__________ measure
_________ measure
IQR =
Q3 - Q1
Range: Max - Min

7. First : Range and IQR = 135 – 45 = 90 = 100-90 = 10 Range = max – min
IQR = Q3 - Q1
= 135 – 45
= 90 =
= 10
Min Q1
Median Q3
Max 45 90 95
100
135

8. First : Range and IQR entire 90 50% middle 10 Range Interpretation:
amount of variation in the_______ sample is _____ mph
entire 90
IQR Interpretation:
amount of variation in the _________ _________
of the sample is ______ mph
middle
50% 10

9. Standard Deviation: Formula
The sample standard deviation is roughly the
average distance between an observation and
the sample mean.
Measures the variability among observed data values.

10. Sample 1 (10 10 10 10 10) no s = ____ Sample: has ____ variation
s = ____ no
Sample: has ____ variation
Mean = 10

11. 2.7: Bell-shaped Distributions and Standard Deviations
The good stuff!
A bell-shaped distribution is a special kind of symmetric distribution.
We care greatly about Normal distributions.
The standard deviation tells us a LOT for a normal distribution.

12. Normal Distribution normal
The ______ distribution is one of the most commonly seen distributions for quantitative data.
A normal distribution has a distinctive symmetric bell shape.

13. Normal Distribution: The empirical rule
Also called the
“ rule”

14. Empirical Rule (E.R.) For a normal distribution…68
_______% of the values (data) fall:
within ± ______ st dev of the mean 1
____% of the values (data) fall:
within ± _____ st dev of the mean 95 2
99.7
_____% of the values (data) fall:
within ± _____ st dev of the mean 3

15. Visualization of the Empirical Rule
Notice: The middle 95% has a width of 4 standard deviations.

16. (for roughly bell-shaped distributions)
Rough way to approximate the standard deviation
(for roughly bell-shaped distributions)
Based on the empirical rule:
• Look at the histogram and estimate the range of the middle 95% of the data.
• The standard deviation is about one fourth of this range.

17. What is the best estimate below of the s.d. for these data?10 20 30 40 50

18. Example Survey Question: “What is your height in inches?
STAT 200 survey results (SP 2016) found:
a normal shape
mean is 67 inches &
st dev is 3 inches.
Range of values
3.7, 4.0, 4.3, 4.6, 4.9, 5.2, 5.5

19. Exercise: Sketch distribution of Actual Heights Mean = 67 inches;
Exercise: Sketch distribution of Actual Heights Mean = 67 inches; StDev = 3 inches

20. Visualize heights distribution with histogram
s = 3 inches

21. Answer Questions using histogram
What percent of Stat 200 students are at most 64 inches in inches in height?
Answer:

22. Answer Questions using histogram
Which percentile
is located 2 standard
deviations below the
mean?
Answer:

23. Answer Questions using histogram
Which height represents
the 84th percentile?
A. 61 inches
B. 64 inches
C. 67 inches
D. 70 inches
E. 73 inches

24. Review: If you understood today’s lecture, you should be able to solve
2.61, 2.87, 2.95, 2.101, 2.103, 2.105, 2.109abc
Recall objectives:
• Use side-by-side boxplots to explore relationship between quantitative and categorical variable
• Understand intuition for two basic measures of spread:
– IQR (interquartile range)
– Standard deviation
• Learn and apply empirical rule for bell-shaped (normal) distributions