Transcript ENGG

Summarising and presenting data
www.anu.edu.au/nceph/surfstat/
Types of data
• Two broad types: qualitative and quantitative
• Qualitative data arise when the observations
fall into separate distinct categories.
Examples are:
Colour of eyes : blue, green, brown etc
Exam result : pass or fail
Socio-economic status : low, middle or high.
•Such data are discrete
Quantitative Data
• Quantitative or numerical data arise when the
observations are counts or measurements
• Discrete if measurements are integers
– number of people in a household,
– number of cigarettes smoked per day
• Continuous if measurements can take any value,
(usually within some range)
– weight
– height
– time
Variables and statistics
•
•
Quantities such as sex and weight are called
variables, because the value of these quantities vary
from one observation to another.
Numbers calculated to describe important features of
the data are called statistics. For example,


the proportion of females
the average age of unemployed persons, in a sample of
residents of a town are statistics.
Example: Commodore data
• Prices of n=38 second-hand cars
6000 6700 3800
20000 11990 16500
9900 18000
9850 9000
11000 9990
7000
10750
5800 9975 10500 5990
9500 12995 12500 8000
9500
9400 7250 15000
5800 29500* 15000 9000
2200
4000 13500 14500
• Continuous
4500 8900
4250 4990
data, need to summarise
Constructing a frequency distribution
• Calculate the range and divide it by the chosen number of intervals
to get the approximate length for each interval.
• Usually use from 5 to 15 intervals.
• Define interval end points so they don't overlap or leave gaps (ie.
they are mutually exclusive and exhaustive) - This ensures that
every observation belongs in exactly one interval.
• It is a usually simpler idea to have all intervals of the same length
• Count the number of values in each interval (the class frequency) go through the data once only and use tally marks to help counting.
• Usually relative frequencies or percentages are helpful to show the
distribution of data.
Frequency distribution
Histogram
• area of rectangle = frequency (or relative frequency)
• But area = length x height
• So if all intervals are the same length, L
Features of a histogram
Mode
• The mode is the value or category which
occurs most frequently.
• If several data values occur with the same
maximal frequency, they are all modes.
• For example, in the Commodore data, using
the grouped data, the class interval, [8,000 10,999], is the modal interval.
Modality and Symmetry
• Modality: No. of peaks
– E.g. one peak-unimodal
• Skewness: departure from symmetry
positive skewness
(skew to the right)
negative skewness
(skew to the left)
Human histogram
Human histogram explained
Process control example
•Deming
•500 steel rods
•Ideal dia. = 1cm
• Is process in control?
• Why the gap?
MEASURES OF CENTRAL
TENDENCY ("Averages")
• Mean (arithmetic mean): x (read as 'x bar')
• Notation: denote data values by x1,x2,…,xn
• n denotes no. of data points
n
x1  x2  ...  xn
x

n
x
i 1
n
i
Mean for frequency distribution
Denote the class midpoints by x1,x2 ,..., xn
Denote the frequencie s by f1,f 2 ,..., f n
Mean for grouped data x 
n
i.e. x   xi f i
i 1
x1 f1  x2 f 2  ...  xn f n
n
Median
• ‘Middle’ value of the data set
• A number which is greater than half the data
values and less than the other half
• (n+1)/2 –th ordered observation
Data set: 6, 6.7, 3.8, 7, 5.8
Ordered: 3.8, 5.8, 6, 6.7, 7
Median: (5+1)/2 ordered obs.
If even: 6, 6.7, 3.8, 7, 5.8, 9.975
Quartiles and percentiles
•
•
•
•
Median: 50% below, 50% above
1st quartile: 25% below, 50% above
Q1: (n+1)/4 ordered observation
Q3 (3rd quartile): (3n+1)/4 ordered
observation
Data set: 6, 6.7, 3.8, 7, 5.8
Ordered: 3.8, 5.8, 6, 6.7, 7
•p-th percentile or quantile: p% below, (100-p)% above
Stem and leaf plot
Finally order the leaves
Percentiles via stem and leaf plot
Get the median:
Median= (n+1)/2 ordered obs.
i.e. 10.5 th ordered observation
Lies in the stem 7|
Median=(72+76)/2 = 74
Get 1st quartile:
Q1 = (n+1)/4 ordered obs.
Get third quartile:
Q3 = (3n+1)/4 ordered obs.
Percentiles from a freq. distr.
•What are median, 1st and 3rd quartiles ?
•Actual values are 6700, 5900 and 10200
•You lose details in a frequency distribution
Comparison of Mean and Median
Data set A: 2,3,3,4,5,7,8
Data set B: 2,3,3,4,5,8,20
Both have n = 7 values.
•The median is not affected by extreme values, but the
mean is changed
•Median is useful for incomplete data
•E.g. consider an experiment to measure average
lifetime of a light bulb (n=6) : 200,400, 650, 700, 900,..
Comparing Mean, Median and Mode
•If distribution is symmetric and
unimodal, all three coincide
•If only symmetric, mean and
median coincide
•If distribution is not symmetric,
better to use median than mean
MEASURES OF VARIABILITY
• Statistics which summarise how spread out the data
values are. Also called measures of dispersion
• The range = max-min (used in quality control)
• The range is susceptible to extreme values
IQR
• The interquartile range is defined as IQR = Q3 - Q1
• IQR is less susceptible to outliers (like the median)
Five number summary
•Boxplot (or box-and-whisker plot)
•Box contains middle 50% of data
•If an obs is > 3 times IQR, it is an outlier
Boxplots are useful for comparing groups
Deviations from the mean
Summarising deviations from mean
The deviation of each value xi from the mean is:
di  xi  x
The mean (or sum) of deviations is not a good summary:
•Instead use a positive function such as di2 or |di|
•Variance or mean square error:
1
•Mean absolute deviation: n  d i
Variance and Standard Deviation
Usually n-1 instead of n is used in the denominator :
sample variance
Problem: squared distances have squared units
s=
the sample standard deviation.
Example: small data set
Data set A: {xi} = 2, 3, 3, 4, 5, 7, 8:
There are n=7 observations and mean = 4.57.
The deviations from the mean, di , are:
-2.57, -1.57, -1.57, -0.57, 0.43, 2.43, 3.43. So
Shortcut formulae for variance
Bivariate methods
• We have (mostly) looked at univariate
methods
• Most interesting problems are bi (or multi)
variate
• Continuous variable vs. qualitative variable:
comparative boxplot
• Continuous variable vs. continuous variable:
scatterplot
Presenting bivariate data
• Scatterplots are useful for illustrating the
relationship between continuous variables
(xi, yi), i = 1,..n
•Indicates type of
relationship
Creating a scatterplot
Student
Height(cm)
weight(kg)
1
167
60
2
170
64
3
160
57
4
152
46
5
157
55
6
160
50
Step 1:
Create variables ht and wt
Step 2:
plot(ht,wt,xlab=“height”,
ylab=“weight”)
Summarising a relationship
plot(temperature,ozone)
abline(lm(ozone~temperature,
data=air))
*
5
*
3
2
*
*
*
*
* *
* * **
* ** *
** * **
*
*
** *
*
*
*
*
*
*
*
*
*
* *
*
**
*
*
* *
*
*
*
* * ** * **
* *
** *
*
*
*
*
*
* **
* *
**
*
*
*
*
*
***
*
*
*
*
*
60
70
80
temperature
*
**
* ** *
*
* * **
*
*
*
* *
*
1
ozone
4
*
90
Summarising a nonlinear
relationship
2
NOx
3
4
•Use a smoother
lines(supsmu(E,NOx))
1
plot(E,NOx)
0.6
0.8
1.0
E
1.2