Statistics 100
Lecture Set 6
Re-cap
• Last day, looked at a variety of plots
• For categorical variables, most useful plots were bar charts and
pie charts
• Looked at time plots for quantitative variables
• Key thing is to be able to quickly make a point using graphical
techniques
Re-cap
• Recall:
– A distribution of a variable tells us what values it takes on and
how often it takes these values.
Histograms
• Similar to a bar chart, would like to display main features of an
empirical distribution (or data set)
• Histogram
– Essentially a bar chart of values of data
– Usually grouped to reduce “jitteriness” of picture
– Groups are sometimes called “bins”
Histogram
8
• Uses rectangles to show number (or percentage) of values in
intervals
4
• X-Axis usually shows intervals
Counts
6
• Y-axis usually displays counts or percentages
0
2
• Rectangles are all the same width
6
8
10
12
Data
14
16
Example (discrete data)
• In a study of productivity, a large number of authors were
classified according to the number of articles they published
during a particular period of time.
Number of
articles
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Frequency
784
204
127
50
33
28
19
19
6
7
6
7
4
4
5
3
3
00
unt
600
800
Example (discrete data)
Example (continuous data)
• Experiment was conducted to investigate the muzzle velocity of
a anti-personnel weapon (King, 1992)
• Sample of size 16 was taken and the muzzle velocity (MPH)
recorded
279.9
294.1
260.7
277.9
285.5
298.6
265.1
305.3
264.9
262.9
256.0
255.3
276.3
278.2
283.6
250.0
Constructing a Histogram – continuous data
• Find minimum and maximum values of the data
• Divide range of data into non-overlapping intervals of equal
length
• Count number of observations in each interval
• Height of rectangle is number (or percentage) of observations
falling in the interval
• How many categories?
Example
• Experiment was conducted to investigate the muzzle velocity of
a anti-personnel weapon (King, 1992)
• Sample of size 16 was taken and the muzzle velocity (MPH)
recorded
279.9
294.1
260.7
277.9
285.5
298.6
265.1
305.3
264.9
262.9
256.0
255.3
276.3
278.2
283.6
250.0
• What are the minimum and maximum values?
• How do we divide up the range of data?
• What happens if have too many intervals?
• Too Few intervals?
• Suppose have intervals from 240-250 and 250-260. In which
interval is the data point 250 included?
Intervals
[240,250)
[250,260)
[260,270)
[270,280)
[280,290)
[290,300)
[300,310)
Frequency
2
1
0
Frequency
3
4
Histogram of Muzzle Velocity
240
260
280
Muzzle Velocity (MPH)
300
320
Interpreting histograms
• Gives an idea of:
– Location of centre of the distribution
– How spread are the data
– Shape of the distribution
• Symmetric
• Skewed left
• Skewed right
• Unimodal
• Bimodal
• Multimodal
– Outliers
•
Striking deviations from the overall pattern
Example – mid-term 1 grades (2011)
• Was out of 34 + a bonus question (n=344)
Example – mid-term 1 grades (2011)
• Too many bins?
Example – mid-term 1 grades (2011)
• Too few bins
Example – mid-term 1 grades (2011)
• Potential outlier?
G r ades
f r om
a St a t s
Cla s s
9 0
1 0 0
8
6
4
2
0
Co u n t
1 0
1 2
1 4
Ex a m
6 0
7 0
8 0
E x a m
1
G
r a d e s
Example
G r ades f or
Un d e r g r a d u a t e s
6
4
2
0
Co u n t
8
1 0
Ex a m
6 0
7 0
8 0
Ex a m
1
9 0
G
r a d e s
1 0 0
Example
G r ades f or
G r a d u a t e St u d e n t s
4
2
0
Co u n t
6
8
Ex a m
6 0
7 0
8 0
Ex a m
1
9 0
G
r a d e s
1 0 0
Numerical Summaries (Chapter 12)
• Graphic procedures visually describe data
• Numerical summaries can quickly capture main features
Measures of Center
• Have sample of size n from some population, x , x ,..., x
1
2
• An important feature of a sample is its central value
• Most common measures of center - Mean & Median
n
Sample Mean
• The sample mean is the average of a set of measurements
• The sample mean:
n
x

x  i1n i
Sample Median
• Have a set of n measurements, x , x ,..., x
1
2
n
• Median (M) is point in the data that divides the data in half
• Viewed as the mid-point of the data
• To compute the median:
1.
For sample size “n”, compute position = (n+1)/2
1.
2.
If position is a whole number, then M is the value at this position of
the sorted data
If position falls between two numbers, then M is the value halfway
between those two positions in the sorted data
Example
• Finding the Median, M, when n is odd
• Example: Data = 7, 19, 4, 23, 18
Muzzle Velocity Example
• Data (n=16)
279.9
294.1
260.7
277.9
285.5
298.6
265.1
305.3
264.9
262.9
256.0
255.3
276.3
278.2
283.6
250.0
Muzzle Velocity Example
• Mean:
279.9  294.1 ... 250.0
x
 274.6
16

Muzzle Velocity Example
• Median:
Sample Mean vs. Sample Median
• Sometimes sample median is better measure of center
• Sample median less sensitive to unusually large or small values
called
• For symmetric distributions the relative location of the sample
mean and median is
• For skewed distributions the relative locations are
Other Measures of interest
• Maximum
• Minimum
• Percentiles
– A percentile of a distribution is a value that cuts off the stated part
of the distribution at or below that value, with the rest at or above
that value.
• 5th percentile: 5% of distribution is at or below this value and 95% is at
or above this value.
• 25th percentile: 25% at or below, 75% at or above
• 50th percentile: 50% at or below, 50% at or above
• 75th percentile: 75% at or below, 25% at or above
• 90th percentile: 90% at or below, 10% at or above
• 99th percentile: ___% at or below, ___% at or above
• Percentiles
– Can be applied to a population or to a sample
• Usually don’t know population
– Use sample percentiles to estimate pop. percentiles
– Standardized tests often measured in percentiles
– Birth statistics often measured in percentiles
• First
–
–
–
daughter
10th percentile weight
25th percentile length
95th percentile head circumference
Important Percentiles
• First Quartile
• Second Quartile
• Third Quartile
Computing the quartiles
• You know how to compute the median
• Q1 =
• Q3 =
Example
• Finding the other quartiles
1.
2.
•
Example: 4, 7, 18, 19, 23,
–
–
•
For Q1, find the median of all values below M.
For Q3, find the median of all values above M.
Q1:
Q3:
Example: 4, 7, 12, 18, 19, 23,
–
–
M=18
Q1:
Q3:
M=15
• 5 number summary often reported:
– Min, Q1, Q2 (Median), Q3, and Max
– Summarizes both center and spread
– What proportion of data lie between Q1 and Q3?
Box-Plot
Displays 5-number summary
graphically
•
Box drawn spanning quartiles
•
Line drawn in box for median
•
Lines extend from box to max.
and min values.
•
Some programs draw whiskers
only to 1.5*IQR above and below
the quartiles
75
70
65
60
Ex am Sc ores
80
85
•
Undergrads
• Can compare distributions
using side-by-side box-plots
• What can you see from the
plot?
Example - Moisture Uptake
•
There is a need to understand degradation of 3013 containers during long term
storage
•
Moisture uptake is considered a key factor in degradation due to corrosion
•
Calcination removes moisture
•
Calcination temperature requirements were written with very pure materials in
mind, but the situation has evolved to include less pure materials, e.g. high in
salts (Cl salts of particular concern)
•
Calcination temperature may need to be reduced to accommodate salts.
•
An experiment is to be conducted to see how the calcination temperature
impacts the mean moisture uptake
Working Example - Moisture Uptake
• Experiment Procedure:
– Two calcination temperatures…wish to compare the mean uptake for each
temperature
– Have 10 measurements per temperature treatment
– The temperature treatments are randomly assigned to canisters
• Response: Rate of change in moisture uptake in a 48 hour period
(maximum time to complete packaging)
Box-Plot
Other Common Measure of Spread:
Sample Variance
• Sample variance of n observations:
n
s 
2
 (x  x)
i 1
i
n 1
•
• Units are in squared units of data
2
Sample Standard Deviation
• Sample standard deviation of n observations:
n
s
•
• Has same units as data
 (x  x)
i 1
i
n 1
2
Exercise
• Compute the sample standard deviation and variance for the
Muzzle Velocity Example
Comments
• Variance and standard deviation are most useful when measure
of center is
• As observations become more spread out, s : increases or
decreases?
• Both measures sensitive to outliers
• 5 number summary is better than the mean and standard
deviation for describing (i) skewed distributions;
(ii) distributions with outliers
Comments
• Standard deviation is zero when
• Measures spread relative to
More interpretation of s
• Empirical Rule
– If a distribution is bell-shaped and roughly symmetric, then
• About 2/3 of data will lie within ±1s of x
• About 95% of data will lie within ±2s of x
• Usually all data will lie within ±3s of x
– So you can reconstruct a rough picture
 of the histogram from just
two numbers


Example
• Mid-term:
Example
• Mid-term:
• Empirical rule tells us