STATISTICS I
COURSE INSTRUCTOR:
TEHSEEN IMRAAN
CHAPTER 4
DESCRIBING DATA
INTRODUCTION
We continue our study of descriptive statistics
with measures of dispersion, such as dot
plots, stem and leaf displays, quartiles,
percentiles, and box plots. Dot plots, a stemand-leaf display, and box plots give additional
insight into where the values are concentrated
and dispersed and the general shape of the
data. Finally we consider bivariate data where
we observe two variables for each individual
or observation selected.
DOT PLOTS
• A graph for displaying a set of data. Each
numerical value is represented by a dot placed
above a horizontal number line.
• To develop a dot plot we display a dot for each
observation along a horizontal number line
indicating the value of each piece of data. For
multiple observations we pile the dots on top
of each other.
STEPS TO CONSTRUCT DOT PLOT
• Sort the data from smallest to largest.
• Draw and label a number line.
• Place a dot . for each observation.
FOR EXAMPLE
Length of Service (in years)
7
6
2
10
6
6
5
8
4
8
4
7
6
5
3
3
7
5
Step 1: Sort the data from smallest to
largest.
2 3 3 4 4 5 5 5 6 6 6 6 7 7 7 8 8 10
Step 2: Draw the number line and
label it as shown.
Step 3: Place a dot for each
observation.
STEM AND LEAF DISPLAYS
• A statistical technique for displaying a set of data. Each
numerical value is divided into two parts: The leading
digit(s) become the stem, and the trailing digits the leaf.
The stems are located along the main vertical axis, and
the leaf for each observation along the horizontal axis.
• To develop a stem-and-leaf chart the first step is to locate
the largest value and the smallest value. This will provide
the range of the stem values. The stem is the leading digit
or digits of the number, and the leaf is the trailing digit.
For example, the number 15 has a stem value of 1 and a
leaf value of 5. For another problem the number 231 has a
stem value of 23 and a leaf value of 1.
FOR EXAMPLE
$12
$28
$32
$24
$17
$6
$34
$18
$22
$42
$36
$26
FOR EXAMPLE
Leading Digit
Trailing Digit
0
6
1
278
2
2468
3
246
4
2
OTHER MEASURES OF DISPERSION
QUARTILES:
– First Quartile The point below which one-fourth
or 25% of the ranked data values lie. (It is
designated Q1 )
– Third Quartile The point below which threefourths or 75% of the ranked data values lie. (It is
designated Q3 )
– Logically the median is the Second Quartile
(designated Q2 ). The values corresponding to Q1 ,
Q2 and Q3 divide a set of data into four equal
parts.
DECILES AND PERCENTILES
• Just as quartiles divide a distribution into 4
equal parts, deciles divide a distribution into
ten equal parts; and percentiles divide a
distribution into 100 equal parts.
• The procedure for finding the quartile, decile,
and a percentile for ungrouped data is to
order the data from smallest to largest. Then
use text formula [4-1].
DECILES AND PERCENTILES
Location of a Percentile,
BOX PLOTS
• A graphical display based on five statistics: the
minimum value, Q1 (the first quartile), Q2 the
median, Q3 (the third quartile) and the
maximum value.
• To construct a box plot we need five pieces of
information. We need the minimum value, Q1
(the first quartile), Q2 the median, Q3 (the third
quartile) and the maximum value.
RELATIVE DISPERSION
• Coefficient of variation: The ratio of the
standard deviation to the arithmetic mean,
expressed as a percent.
FORMULA FOR CV
Coefficient of
Variation,
Multiplying by
100 converts the
[4-2]
decimal to a
percent
COEFFICIENT OF VARIATION
• Characteristics of the coefficient of variation
are:
– It reports the variation relative to the mean.
– It is useful for comparing distributions with
different units.
SKEWNESS
• Four shapes of distribution
• Coefficient of skewness: A measure to
describe the degree of skewness. How the
distribution is skewed?
Text Formula [4–3] is for Pearson’s
Coefficient of Skewness.
Characteristics of the coefficient of
skewness are:
• The coefficient of skewness, designated sk,
measures the amount of skewness and may
range from -3.0 to +3.0.
• A value near -3, such as -2.57, indicates
considerable negative skewness.
• A value such as 1.63 indicates moderate positive
skewness.
• A value of 0, which will occur when the mean and
median are equal, indicates the distribution is
symmetrical and that there is no skewness.
SUMMARY OF CHARTS
RELATIONSHIP BETWEEN TWO
VARIABLES
• Bivariate data: A collection of paired data values.
• Scatter diagram: A graph in which paired data
values are plotted on an X,Y Axis.
• The steps to follow in developing a scatter
diagram are:
– We need two variables.
– We scale one variable (x) along the horizontal axis (X –
Axis) of a graph and the corresponding variable (y)
along the vertical axis (Y – Axis).
– Place a dot for each (x, y) pair of observations.
GRAPH
CONTINGENCY TABLE
• A table used to classify sample observations
according to two or more identifiable
characteristics.
• When we study the relationship between two
or more variables when one or both are
nominal or ratio scale, we tally the results into
a two-way table. This two-way table is
referred to as a contingency table.
CONTINGENCY TABLE
Gender
Bought
Lunch
Boys
Girls
Total
0 up to 10
10
5
15
10 up to 20
20
25
45
Total
30
30
60
CONTINGENCY TABLE
• A contingency table is a cross tabulation that
simultaneously summarizes two variables of
interest and their relationship.
• A survey of 60 school children classified each
as to gender and the number of times lunch
was purchased at school during a four-week
period. Each respondent is classified according
to two criteria – the number of times lunch
was purchased and gender.