Descriptive Statistics Graphing Techniques

Points and grades from examination

Sample size n=30
Data sorting → Frequency table
both for quantitative and qualitative data

Exam grade

Notation
Frequency … ni
Relative frequency … fi
Cumulative Frequency … Ni
Cumulative Percent … Fi

Points from class test

Quantitative variables
Grouping into class intervals

How to select the intervals
Number of intervals → in order to describe the characteristics of the data
Simple reccommendation
intervals of the same width
k … number of intervals
n … sample size

…then
h … width of interval
R … Range=xmax-xmin
k … number of intervals
Our example:
n=30
R=20-2=18

Points from class test

Measures of Central Tendency
Measures that represent
with a proper value the tendency of most data to gather around this value
Number of different measures of central tendency
the arithmetic mean
the median
the mode

The arithmetic mean
Notation
arithmetic mean ……
the sum of the values of a variable divided by the number of scores (by the sample size)

Properties of the arithmetic mean
it is expressed in the same unit of measure as the observed variable
it is the point in a distribution of measurements about which the sum of deviations are equal to zero
Note: deviation explains the distance and direction from
a reference point – here the arithmetic mean, it is positive when the value is greater than the mean and negative when lower than the mean
3. the mean is very sensitive to extreme values

Personal income (thousands CZK)
thousands CZK

12 of 16 values are below the arithmetic mean, because of the highest value x16=120,5 (directors income)
personal income is a commonly studied variable in which other measure of central tendency is preferred

Other measures of central tendency
The median….
The value above and below which one-half of the frequencies fall
n…odd number
median case number=(n+1)/2
n…even number
the arithmetic mean of the two middle values
Properties: Insensitive to extreme values

Other measures of central tendency
The mode….
The value that occurs with greatest frequency
for qualitative (nominal and ordinal) and quantitative discrete data
from a statistical perspective it is also the most probable value

Personal income (thousands CZK)
n=16… even number
the median the mode

Personal income (thousands CZK)
n=16… even number
the median the mode

Use of mean, median and mode
The arithmetic mean
member of mathematical system in advanced statistical analysis
preferred measure of central tendency if the distribution is not skewed
The median
when the distribution is skewed
The mode
whenever a quick, rough estimate of central tendency is desired

The mean, median, mode and skewness

to describe the spread of the data, its variation around a central value
we want to express the distance along the scale of values
Measures of Dispersion

The Range….R
it is the distance between the largest and the smallest value
R=xmax-xmin
it does not explain the variability inside the range !
very simple and straightforward measure of dispersion

The Variance…s2
it is an average squared deviation of each value from the mean
it is the sum of the squared deviations from the mean divided by n
when computing the variation based on sample we correct the calculation

Working formulas
For easier computation
Formula 1
Formula 2

the variance explains both
the variability of the values around the arithmetic mean
the variability among the values
difficult interpretation
(it is expressed in the squares of the unit of measure)

The Standard Deviation…s
it is the square root of variance
when computing the variation based on sample

it is expressed in the same unit of measure as the observed variable
the size of the standard deviation is related to the variability in the values
the more homogeneous values, the smaller SD
the heterogeneous values, the larger SD
member of mathematical system in advanced statistical analysis (like the arthmetic mean)
Properties of the standard deviation

Two data sets with the same arithmetic mean and different SD

Example – Personal income (thousands CZK)
thousands CZK

Coefficient of Variation…V
the ratio of the standard deviation to the mean
often reported as a percentage (%) by multiplying by 100

it is a relative measure of dispersion
used when comparing two data sets with different units or widely different means
values higher than 50% indicate large variability

Example – Personal income (thousands CZK)

Percentiles (Centiles)
value below which a certain percent of observations fall
scale of percentile ranks is comprised of 100 units
insensitive to extreme values

Deciles
divides a distribution into 10 equal parts
there are 9 deciles
D1 – 1st decile
- 10 percent of values fall below it
D9 – 9th decile
- 90 percent of values fall below it

divides a distribution into 4 equal parts
Q1 - 25 percent of values fall below it
- 25th centile
Q2 - 50 percent of values fall below it
- 50th centile
Q3 – 75 percent fall below it
- 75th centile
Quartiles

Graphing Techniques

Constructing graphs – Bar graph
x – axis: labels of categories
y – axis: frequency (relative frequency)
The height of each rectangle is the category`s frequency or relative frequency.

Arranging the graph
nominal variables – we can arrange the categories in any order:alphabetically, decreasing/increasing order of frequency
ordinal variables – the categories should be placed in their naturally occuring order

Constructing graphs – Pie graph
Pie chart – a circle divided into sectors
each sector represents a category of data
the area of each sector is proportional to the frequency of the category

Constructing graphs – Histogram
bar graph for quantitative data
values are grouped into intervals (classes)
constructed by drawing rectangles for each class of data
the height of each rectangle is the frequency of the class
the width of each rectangle is the same

Histogram

Constructing graphs – Boxplot
box-and-whisker diagram
five number summary

Boxplot
Q3
Q1
Q2