Lect 3 background mathematics

Know your Data
(Background Mathematics)
DATA MINING AND
DATA WAREHOUSING
CS- 6301 (Cr-3)
Dr. Hrudaya Kumar Tripathy

Basic Statistical Descriptions of Data
The entire subject of statistics is based around the idea that we have this
big set of data, and we want to analyse that set in terms of the
relationships between the individual points in that data set.
• Motivation
• To better understand the data: central tendency, variation and spread
• Data dispersion characteristics
• median, max, min, quantiles, outliers, variance, etc.
• Numerical dimensions correspond to sorted intervals
• Data dispersion: analyzed with multiple granularities of precision
• Boxplot or quantile analysis on sorted intervals
• Dispersion analysis on computed measures
• Folding measures into numerical dimensions
• Boxplot or quantile analysis on the transformed cube

Measuring the Central Tendency
• The most common and effective numeric
measure of the “center” of a set of data is
the (arithmetic) mean. Let x1,x2, ...... ,xN be
a set of N values or observations, such as
for some numeric attribute X, like salary.
• Sometimes, each value xi in a set may be
associated with a weight wi for i = 1, .... ,N.
The weights reflect the significance,
importance, or occurrence frequency
attached to their respective values. In this
case, we can compute
Mean (algebraic measure) (sample vs. population):
Note: N is population size.
The mean of this set of values is
This is called the weighted arithmetic
mean or the weighted average.

• A trimmed mean (sometimes called a truncated mean) is similar to a mean, but it trims any
outliers. Outliers can affect the mean (especially if there are just one or two very large values), so
a trimmed mean can often be a better fit for data sets with erratic high or low values or for
extremely skewed distributions. Even a small number of extreme values can corrupt the mean.
• For example, the mean salary at a company may be substantially pushed up by that of a few
highly paid managers. Similarly, the mean score of a class in an exam could be pulled down quite a
bit by a few very low scores.
• Which is the mean obtained after chopping off values at the high and low extremes.
• Example: Find the trimmed 20% mean for the following test scores: 60, 81, 83, 91, 99.
• Step 1: Trim the top and bottom 20% from the data. That leaves us with the middle three
values: 60, 81, 83, 91, 99.
• Step 2: Find the mean with the remaining values. The mean is (81 + 83 + 91) / 3 ) = 85.

• Median:
• The median of a set of data is the middlemost number in the set. The
median is also the number that is halfway into the set.
• To find the median, the data should first be arranged in order from
least to greatest.
• Middle value if odd number of values, or average of the middle two
values otherwise
• What will be the median estimated by interpolation (for grouped
data)?

• Mode
• The mode for a set of data is the value that occurs most frequently in the set.
Therefore, it can be determined for qualitative and quantitative attributes.
• It is possible for the greatest frequency to correspond to several different
values, which results in more than one mode.
• Data sets with one, two, or three modes are respectively called unimodal,
bimodal, and trimodal.
• In general, a data set with two or more modes is multimodal.
• At the other extreme, if each data value occurs only once, then there is no
mode.
• Example:
• 53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70
• 62 appears three times, more often than the other values, so Mode = 62

Mean, Median and Mode from Grouped Frequencies
The Race.....This starts with some raw data (not a grouped frequency
yet) ...
To find the Mean Alex adds up all the numbers, then divides by how many numbers:
Mean = (59+65+61+62+53+55+60+70+64+56+58+58+62+62+68+65+56+59+68+61+67)/21
= 61.38095...

• To find the Median Alex places the numbers in value order and finds the
middle number.
• In this case the median is the 11th number:
53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70
• Median = 61
• To find the Mode, or modal value, Alex places the numbers in value order
then counts how many of each number. The Mode is the number which
appears most often (there can be more than one mode):
53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70
• 62 appears three times, more often than the other values, so Mode = 62

• Grouped Frequency Table
• Alex then makes a Grouped Frequency Table:
• So 2 runners took between 51 and 55 seconds, 7 took
between 56 and 60 seconds, etc

We can estimate the Mean by using the midpoints.
Let's now make the table using midpoints:
Estimating the Mean from Grouped Data

Estimating the Mean from Grouped Data
• Our thinking is: "2 people took 53 sec, 7 people took 58
sec, 8 people took 63 sec and 4 took 68 sec". In other
words we imagine the data looks like this:
53, 53, 58, 58, 58, 58, 58, 58, 58, 63, 63, 63, 63, 63, 63, 63, 63, 68, 68, 68, 68
• Then we add them all up and divide by 21. The quick way
to do it is to multiply each midpoint by each frequency:
And then our estimate of the mean time to complete the
race is:

Estimating the Median from Grouped Data
• Let's look at our data again:
• The median is the middle value, which in our
case is the 11th one, which is in the 61 - 65
group:
• We can say "the median group is 61 - 65"
• But if we want an estimated Median value we
need to look more closely at the 61 - 65 group.

• At 60.5 we already have 9 runners, and by the next
boundary at 65.5 we have 17 runners.
• By drawing a straight line in between we can pick out
where the median frequency of n/2 runners is:
Estimating the Median from Grouped Data
And this handy formula does the calculation:
where:
L is the lower class boundary of the group
containing the median
n is the total number of values
B is the cumulative frequency of the groups before
the median group
G is the frequency of the median group
w is the group width

For our example:
L = 60.5
n = 21
B = 2 + 7 = 9
G = 8
w = 5
width
freq
lfreqn
Lmedian
median
)
)(2/
(1



Estimating the Mode from Grouped Data
• Again, looking at our data:
• We can easily find the modal group (the group with the highest
frequency), which is 61 - 65
• We can say "the modal group is 61 - 65"
• But the actual Mode may not even be in that group! Or there may
be more than one mode. Without the raw data we don't really
know. But, we can estimate the Mode using the following
formula:
where:
L is the lower class boundary of the modal group fm is the frequency of the modal
group
fm-1 is the frequency of the group before the modal group w is the group width
f is the frequency of the group after the modal group

L = 60.5
fm-1 = 7
fm = 8
fm+1 = 4
w = 5
For our example:

Baby Carrots Example
Example: You grew fifty baby carrots using
special soil. You dig them up and measure
their lengths (to the nearest mm) and
group the results:
Age Example
The ages of the 112 people who live on
a tropical island are grouped as follows:

Symmetric vs. Skewed Data
• Median, mean and mode of symmetric, positively and negatively skewed data
• Data in most real applications are not symmetric (a). They may instead be either
positively skewed (b), where the mode occurs at a value that is smaller than the
median or negatively skewed (c), where the mode occurs at a value greater than
the median.

Measuring the Dispersion of Data
• Range, Quartiles, Variance, Standard Deviation, and Interquartile Range
• We now look at measures to assess the dispersion or spread of numeric data.
The measures include range, quartiles, percentiles, and the interquartile
range.
• The five-number summary, which can be displayed as a boxplot, is useful in
identifying outliers.
• Variance and standard deviation also indicate the spread of a data
distribution.

Range
• Let x1, x2, . . . , xN be a set of observations for some numeric attribute, X.
• The range of the set is the difference between the largest (max()) and smallest (min())
values.
Standard Deviation
• The standard deviation (usually abbreviated SD, sd, or just s) of a bunch of numbers tells
you how much the individual numbers tend to differ (in either direction) from the mean.
It’s calculated as follows:
Measuring the Dispersion of Data (cont..)
This formula is saying that you calculate the
standard deviation of a set of N numbers (Xi) by
subtracting the mean from each value to get the
deviation (di) of each value from the mean,
squaring each of these deviations, adding up the
(di)2 terms, dividing by N – 1, and then taking the
square root.

Standard Deviation (cont.)
• This is almost identical to the formula for the root-
mean-square deviation of the points from the mean,
except that it has N – 1 in the denominator instead of
N.
• This difference occurs because the sample mean is
used as an approximation of the true population
mean (which you don’t know). If the true mean were
available to use, the denominator would be N.
• When talking about population distributions, the SD
describes the width of the distribution curve. The
figure shows three normal distributions. They all
have a mean of zero, but they have different
standard deviations and, therefore, different widths.
Each distribution curve has a total area of exactly 1.0,
so the peak height is smaller when the SD is larger.
For an IQ example (84, 84, 89, 91, 110, 114, and 116)
where the mean is 98.3, you calculate the SD as follows:
Standard deviations are very sensitive to extreme
values (outliers) in the data. For example, if the highest
value in the IQ dataset had been 150 instead of 116,
the SD would have gone up from 14.4 to 23.9.

Why n-1 in Standard Deviation?
Why divide by n-1 rather than n?
You compute the difference between each value and the mean of those values. You don't
know the true mean of the population; all you know is the mean of your sample. Except
for the rare cases where the sample mean happens to equal the population mean, the
data will be closer to the sample mean than it will be to the true population mean.
So the value you compute will probably be a bit smaller (and can't be larger) than what it
would be if you used the true population mean.
To make up for this, divide by n-1 rather than n. This is called Bessel's correction.
But why n-1? If you knew the sample mean, and all but one of the values, you could
calculate what that last value must be. Statisticians say there are n-1 degrees of freedom.
Bessel's correction

• Several other useful measures of dispersion are related to the SD:
• Variance: Variance of N observations, x1, x2, . . . , xN, for a numeric attribute X
is
• The variance is just the square of the SD. For the IQ example, the variance =
14.42 = 207.36.
• Coefficient of variation: The coefficient of variation (CV) is the SD divided by
the mean. For the IQ example, CV = 14.4/98.3 = 0.1465, or 14.65 percent.

Quartiles
It divide an ordered data set into four equal parts.
The values which divide each part are called the first, second, and third quartiles; they are
denoted by Q1, Q2, and Q3, respectively.
Q1 is the middle value of the first half of the ordered data set, Q2 is the median value in
the set, and Q3 is the middle value in the second half of the ordered data set.
Interquartile range (IQR)
A good measure of the spread of data is the interquartile range (IQR) or the difference
between Q3 and Q1. This gives us the width of the box, as well. A small width means more
consistent data values since it indicates less variation in the data or that data values are
closer together. So, IQR = Q3 - Q1

Interquartile Range = Q3-Q1
• With an Even Sample Size:
• For the sample (n=10) the median
diastolic blood pressure is 71 (50% of the
values are above 71, and 50% are below).
• The quartiles can be determined in the
same way we determined the median,
except we consider each half of the data
set separately.
• With an Odd Sample Size:
• For the sample (n=10) the median
diastolic blood pressure is 72 (50% of the
values are above 72, and 50% are below).
• When the sample size is odd, the median
and quartiles are determined in the same
way.
• Suppose in the previous example, the
lowest value (62) were excluded, and the
sample size was n=9. The median and
quartiles are indicated below.

Outliers and Tukey Fences:
• Tukey Fences
• When there are no outliers in a sample, the mean and standard deviation are used to
summarize a typical value and the variability in the sample, respectively.
• When there are outliers in a sample, the median and interquartile range are used to
summarize a typical value and the variability in the sample, respectively.
• Outliers are values below Q1-1.5(Q3-Q1) or above Q3+1.5(Q3-Q1) or equivalently, values
below Q1-1.5 IQR or above Q3+1.5 IQR.
• In previous example, for the diastolic blood pressures, the lower limit is 64 - 1.5(77-64) =
44.5 and the upper limit is 77 + 1.5(77-64) = 96.5. The diastolic blood pressures range from
62 to 81. Therefore there are no outliers.

Example : The Full Framingham Cohort Data
• The Framingham Heart Study is a long-term, ongoing cardiovascular cohort study on residents
of the city of Framingham, Massachusetts. The study began in 1948 with 5,209 adult subjects
from Framingham, and is now on its third generation of participants.
• Table 1 displays the means, standard deviations, medians, quartiles and interquartile ranges for
each of the continuous variables in the subsample of n=10 participants who attended the
seventh examination of the Framingham Offspring Study.
Table 1 - Summary Statistics on n=10 Participants

• Table 2 displays the observed minimum and maximum values along with the limits to determine
outliers using the quartile rule for each of the variables in the subsample of n=10 participants.
• Are there outliers in any of the variables? Which statistics are most appropriate to summarize
the average or typical value and the dispersion?
Table 2 - Limits for Assessing Outliers in Characteristics Measured in the n=10 Participants
Since there are no suspected
outliers in the subsample of n=10
participants, the mean and
standard deviation are the most
appropriate statistics to
summarize average values and
dispersion, respectively, of each
of these characteristics.

Continue.....
• For clarity, we have so far used a very small subset of the Framingham Offspring Cohort to
illustrate calculations of summary statistics and determination of outliers. For your interest,
Table 3 displays the means, standard deviations, medians, quartiles and interquartile ranges
for each of the continuous variable displayed in Table 1 in the full sample (n=3,539) of
participants who attended the seventh examination of the Framingham Offspring Study.
Table 3-Summary Statistics on Sample of (n=3,539) Participants

• Table 4 displays the observed minimum and maximum values along with the limits to determine
outliers using the quartile rule for each of the variables in the full sample (n=3,539).
Continue.....
Table 4 - Limits for Assessing Outliers in Characteristics Presented in Table 3
Are there outliers in any of
the variables?
Which statistics are most
appropriate to summarize
the average or typical values
and the dispersion for each
variable?

• In the full sample, each of the characteristics has outliers on the upper end of the distribution
as the maximum values exceed the upper limits in each case. There are also outliers on the low
end for diastolic blood pressure and total cholesterol, since the minimums are below the lower
limits.
• For some of these characteristics, the difference between the upper limit and the maximum (or
the lower limit and the minimum) is small (e.g., height, systolic and diastolic blood pressures),
while for others (e.g., total cholesterol, weight and body mass index) the difference is much
larger. This method for determining outliers is a popular one but not generally applied as a hard
and fast rule. In this application it would be reasonable to present means and standard
deviations for height, systolic and diastolic blood pressures and medians and interquartile
ranges for total cholesterol, weight and body mass index.
Observations on example......

Boxplot Analysis
• Boxplots are a popular way of visualizing a distribution. A
boxplot incorporates the five-number summary as follows:
• Five-number summary of a distribution
• Minimum, Q1, Median, Q3, Maximum
• Boxplot
• Data is represented with a box
• The ends of the box are at the first and third quartiles, i.e.,
the height of the box is IQR
• The median is marked by a line within the box
• Two lines (called whiskers) outside the box extend to the
smallest (Minimum) and largest (Maximum) observations.
• Outliers: points beyond a specified outlier threshold,
plotted individually
Boxplot for the unit price data for items sold at four branches of
AllElectronics during a given time period.

Finally, boxplots often provide information about the shape of a data set. The examples below
show some common patterns.
Boxplot Analysis (cont..)
Each of the boxplots illustrates a different
skewness pattern.
If most of the observations are concentrated on
the low end of the scale, the distribution is skewed
right; and vice versa.
If a distribution is symmetric, the observations will
be evenly split at the median, as shown in the
middle figure.

The beauty of the normal curve:
68-95-99.7 Rule
No matter what  and  are,
• the area between - and + is about 68%;
the area between -2 and +2 is about
95%;
• and the area between -3 and +3 is
about 99.7%.
• Almost all values fall within 3 standard
deviations. (μ: mean, σ: standard deviation)

Are my data “normal”?
 Not all continuous random variables are normally distributed!!
 It is important to evaluate how well the data are approximated by
a normal distribution

Quantile Plot
• It cover common graphic displays of data distributions.
• A quantile plot is a simple and effective way to have a first look at a univariate data
distribution. It used to check whether your data is Normal.
• Let xi, for i = 1 to N, be the data sorted in increasing order so that x1 is the smallest
observation and xN is the largest for some ordinal or numeric attribute X.
• Each observation, xi, is paired with a percentage, fi, which indicates that approximately
fi × 100% of the data are below the value, xi.
• We say “approximately” because there may not be a value with exactly a fraction, fi, of
the data below xi.
• Note that the 0.25 percentage corresponds to quartile Q1, the 0.50 percentage is the
median, and the 0.75 percentage is Q3.

On a quantile plot, xi is graphed against fi. This allows us to compare different
distributions based on their quantiles. For example, given the quantile plots
of sales data for two different time periods, we can compare their Q1,
median, Q3, and other fi values at a glance.
Quantile Plot (cont..)

Quantile–Quantile Plot
• A quantile–quantile plot, or q-q plot, graphs the quantiles of one univariate
distribution against the corresponding quantiles of another.
• It is a powerful visualization tool in that it allows the user to view whether there
is a shift in going fromone distribution to another.
Suppose that we have two sets of observations for the attribute or
variable unit price, taken from two different branch locations.
• Let x1, .... ,xN be the data from the first branch, and y1, ......, yM be the
data from the second, where each data set is sorted in increasing
order.
• If M = N (i.e., the number of points in each set is the same), then we
simply plot yi against xi , where yi and xi are both (i-0.5)/N quantiles
of their respective data sets.
• If M < N (i.e., the second branch has fewer observations than the
first), there can be only M points on the q-q plot. Here, yi is the (i-
0.5)/M quantile of the y data, which is plotted against the (i -0.5)/M
quantile of the x data. This computation typically involves
interpolation.

Each point corresponds to the same quantile for each data set and shows the unit price of items sold at branch
1 versus branch 2 for that quantile.
Quantile–Quantile Plot (cont.)
(To aid in comparison, the straight line represents the
case where, for each given quantile, the unit price at
each branch is the same.
• The darker points correspond to the data for Q1,
the median, and Q3, respectively.)
• For example, that at Q1, the unit price of items sold
at branch 1 was slightly less than that at branch 2.
• In other words, 25% of items sold at branch 1 were
less than or equal to $60, while 25% of items sold
at branch 2 were less than or equal to $64. At the
50th percentile (marked by the median, which is
also Q2), we see that 50% of items sold at branch 1
were less than $78, while 50% of items at branch 2
were less than $85.
In general, we note that there is a shift in the distribution of branch 1 with respect to branch 2 in that the unit prices of items
sold at branch 1 tend to be lower than those at branch 2.

40
Scatter Plots and Data Correlation
• Provides a first look at bivariate data to see clusters of points, outliers, etc
• Each pair of values is treated as a pair of coordinates and plotted as points in the plane

• The scatter plot is a useful method for providing a first look at bivariate data
to see clusters of points and outliers, or to explore the possibility of
correlation relationships.
• Two attributes, X, and Y, are correlated if one attribute implies the other.
Correlations can be positive, negative, or null (uncorrelated).
• If the plotted points pattern slopes from lower left to upper right, this
means that the values of X increase as the values of Y increase, suggesting a
positive correlation .
• If the pattern of plotted points slopes from upper left to lower right, the
values of X increase as the values of Y decrease, suggesting a negative
correlation.
• A line of best fit can be drawn to study the correlation between the
variables.
Scatter Plots and Data Correlation (cont.)

42
Positively and Negatively Correlated Data
• The left half fragment is positively
correlated
• The right half is negative correlated

44
Histogram Analysis
• Histogram: Graph display of tabulated
frequencies, shown as bars
• It shows what proportion of cases fall into
each of several categories
• Differs from a bar chart in that it is the area of
the bar that denotes the value, not the height
as in bar charts, a crucial distinction when the
categories are not of uniform width
• The categories are usually specified as non-
overlapping intervals of some variable. The
categories (bars) must be adjacent
0
5
10
15
20
25
30
35
40
10000 30000 50000 70000 90000

45
Histograms Often Tell More than Boxplots
 The two histograms shown in the left
may have the same boxplot
representation
 The same values for: min, Q1,
median, Q3, max
 But they have rather different data
distributions

Lect 3 background mathematics

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