STAT 2040 Lecture 2: Jan 16-Jan 20

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20 Jan 2017
Department
Course
Professor
Descriptive Statistics
In descriptive statistics, plots and numerical summaries are used to describe a data set.
Plots for Categorical Variables
Acategorical variable is a variable that falls into one of two or more categories.
Examples: A university student’s major, the province in which a Canadian resides, a per-
son’s blood type.
We illustrate the distribution of a categorical variable by displaying the count or proportion
of observations in each category. We might use bar charts, Pareto diagrams (an ordered bar
chart) or pie charts.
Example. Gun calibre for buyback guns and guns used in homicides and suicides. (From
a buyback program in Milwaukee.)
Gun Calibre Buybacks Homicides Suicides
Small 719 75 40
Medium 182 202 72
Large 20 40 13
Other 20 52 0
Small Medium Large Other
Frequency
050 100 150 200
(a) Bar chart for the homicide guns.
Small
Medium
Large
Other
(b) Pie chart for the homicide guns.
Buybacks Homicides Suicides
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Relative Frequency
Other
Large
Medium
Small
(c) Side-by-side bar charts.
Buybacks Homicides Suicides
0.0
0.2
0.4
0.6
0.8
1.0
Relative Frequency
Other
Large
Medium
Small
(d) Stacked bar charts.
Do excuses 'it
Chapter 3
Area is the
proportion
I
Count
riron
Prop
°
Frequency
n
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Plots for Quantitative Variables
A quantitative variable is a numeric variable that represents a measurable quantity. Exam-
ples: the height of a student, the length of stay in hospital after a certain type of surgery,
the weight of a newborn African elephant.
For quantitative variables, numerical summaries such as averages have meaning.
To illustrate the distribution of a quantitative variable, we plot the dierent values of the
variable and how often these values occur. This can be done in a variety of ways, including:
histograms, stem plots, boxplots, and dot plots.
To create a histogram, we first create a frequency table. In a frequency table, a quan-
titative variable is divided into a number of classes (also known as bins), and the class
boundaries and frequency of each class is listed.
A histogram is a plot of the class frequencies, relative frequencies, or percent relative fre-
quencies against the class boundaries (or class midpoints).
Histogram of survival times (days) for 60 guinea pigs injected with tubercle bacilli:
Lifetime
Frequency
100 200 300 400 500 600
0510 15
Histograms and stemplots allow us to see the distribution of the data (what values the
variable takes on, and how often it takes on these values).
red
µWg'
gung -lstemdeaf displays)
fy
g
28-30
Something sto take note of "
-where is It centred ?
(we 'll use the means
median as measures of
Centre )
-How much variability is there ?
1we will use the Variance Estandard deviation as
--
measures of variability
-Are there any outliers ?(extreme values )
-what is the Shape ?(important )
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Some common distribution shapes.
(a) A symmetric distribution.
Frequency
1000 2000 3000 4000
020 40 60 80
(b) A bimodal distribution.
(a) A distribution that is skewed to the right. (b) A distribution that is skewed to the left.
Figures illustrating right and left skewness.
Lifetime
Frequency
100 200 300 400 500 600
0510 15
(a) Lifetimes of 60 guinea pigs. Time-to-event data
is often right skewed.
1000 2000 3000 4000 5000
050 100 150
Weight(grams)
Frequency
(b) Birth weights of a random sample of 1000
Canadian male births in 2009.
Real world data sets illustrating right and left skewness.
-This is : -roughly symmetric
-Uni model (one peak )
-approx . normal (belt -shaped )
#
alternative
shapes
-not symmetric -AKA :Negatively
-right skew is
III.FEET :\ )skewed
-often seen in :time
to event data ,
Incomes ,housing
prices ,÷
ikjeytedtoiu -#IIETII ..
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