Summary Mathematical Statistics 214

Week 1: 13-17 Feb
Descriptive and exploratory statistics


leaves a tree > e.g. leaf width
POPULATION
, e.g. of




Any of
set
objects, individuals or items with respectto the phenomenon or characteristic thatis

studied, and which forms the whole the objects thatare being studied
or
totality of


>one whole tree


·
defined
by a seto f descriptive conditions

·
has a finite / infinite number of elements



variable:

an observable characteristic a
of
population;e.g. leaf width (random variable X)

be
can
qualitative or
quantative
↓

defined by specifying a


no. of categories:descriptive (observed, notmeasured)




discreet:fixed, isolated values (1, 2, 3, 4 ...)
Quantative

↓
continuous:variable can take on all
possible values within a specific, logical
interval (e.g. age, weight)
amounts thatcan

be counted or measured



measurementof a characteristic

. can be on a nominal, ordinal, interval or ratio scale

-
Nominal:

grouping items into a number categories
of
(only qualitative), e.g. male/female, names,
-
Ordinal: eye colors

ifthe categories can be ordered (e.g. bad, medium, good
-
Interval: e.g. temperature
~>

size the
of difference between any consecutive values (numeric infinitely many values)
e.g. age, temperature, la test

-
RATO?

measurementon the interval scale, butan absolute zero
pointexists
e.g. Kelvin measurements; 100 Kelvin is double so Kelvin;weight, height, distance

, A D

Qualitative Quantative


SAMPLE

subseto felements of the population that is selected to be examined
any
random sample of size ofleaves
e.g.

sampling process -



selecting
of a sample from the population
-
used to make conclusions aboutthe population as a whole

methods
sampling
· random

· stratifiedrandom

·
systematic
· cluster



NOtatiOR

·
population: X random variable
=




leaf-width
-
all
of leaves in particular tree




X -
n(x,08) sample:
E(x) 4
3,5, 3,79...xm 5,53
=



x1 x2
=
= =




var(x) 5 3
0,8 4,07 and
0,72
=
= =

-

,frequency

>
frequency tables 5
A
Discreetn o. of observations
B 3
-
for each value.
for discreetdata: [ 2




grouped frequency table

for continuous data:

Need to items
*
group
into classes




tables for continuous data
frequency

·
Need to
specify intervals for each ofthe classes

range
-




-
no, intervals

-interval widths
-interval limits
-




frequencies

① Range r
largest
=
-

smallestvalue


⑥ No, intervals Typically 6


sturges' rule:1 = 1 +
3.310gion
↑ ↑
no class intervals total no. Observations

③ Interval width preferably equal width

convenient/ sensible

④Internal limits Include all observations


⑥ frequencies No. occurences
of a
of
particular event.

↳




> Interval width
frequencies
-range
no, intervals
12,3] (3,43 (4,5]

(2, 33, interval width is I I 9
(5,6]
10
(3,4],
=>

e.g.

, HISTOGRAM (continuous data)
classification I
grouping
· visual representation
can lead to a loss of

information. (No exact
· area corresponds to
frequency
values of individual items)
· have
equal length intervals
heighta frequencies
->




· bars touch

Frequency distribution: ·
NB:area
corresponds to
frequency crelative

/ data that Area b h frequency)
graph set x
=




shows frequencies then the
equal width,
how ·
If intervals are of
not

are distributed.
the
class
frequencies are not
proportional to



rectangle heights.
CHART
PIE

· Area & Proportion I percentages HEAT MAP

·

Represented on a 360° ·
color indicates observed values (percentages)
circle divided into sectors ·
proportion a color



BAR CHARTS <discreetdata)

·
graphical representation discrete
of and
tables
categorical frequency
·
bars are
separate

FREQUENCY POLYGON

·
figure obtained when heights histogram rectangles
of are
plotted against
midpoints class
of intervals
·
an additional interval is added each
at side (same width) with a
frequency of

o to "anchor" the graph
·
can compare multiple samples STEM AND LEAF DISPLAY

·
similar shape to histogram
CONTINGENCY TABLES ·
Stem:basic interval

·

frequency table for multiple variables ·
leaves:individual observations

2-way tables e.g. 28
-




01076
multiway tables
i
-




00197
·
summaries differenttypes variables
of 15

Stem

fading Dead
Healthy be rounded
· observations can

width < 4 L 0 I
· Stem can be in 19/109/100s etc.
width 4 I 2 2