Recap: List of Topics
Descriptive Statistics
Probability Theory
Inferential Statistics
Different types of data
Different types of plots
Measures of centrality and spread
Sample spaces, events, axioms
Discrete and continuous RVs
Bernoulli, Uniform, Normal dist.
Sampling strategies
Interval Estimators
Hypothesis testing (z-test, t-test)
ANOVA, Chi-square test
Linear Regression
What are the different measures of centrality?
What are some characteristics of these measures?
Learning Objectives
What do the measures of centrality look like for different types of distributions?
How do you compute these measures from histograms?
Why do we need measures of spread and centrality?
What is the effect of certain transformations on these measures?
What are the different measures of centrality?
Why do we need measures of centrality and spread?
Motivation: Summarise Big Data
| Age | Height | Weight | Cholesterol | Sugar level | .... ..... |
|---|---|---|---|---|---|
| 32 | 165 | 75 | 124 | 108 | ... |
| 24 | 172 | 81 | 112 | 98 | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
Imagine a million such rows!
Drawing plots can give a good visual summary
In some situations, we want an even more succinct summary (say, a single/few numbers)
Motivation: Summarise Big Data
| Age | Height | Weight | Cholesterol | Sugar level | .... ..... |
|---|---|---|---|---|---|
| 32 | 165 | 75 | 124 | 108 | ... |
| 24 | 172 | 81 | 112 | 98 | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
A parameter is any numeric property of the entire population under study
A statistic is any numerical property of the sample of a population (used as an estimate for the corresponding parameter of the population)
Recall
Motivation: Summarise Big Data
| Age | Height | Weight | Cholesterol | Sugar level | .... ..... |
|---|---|---|---|---|---|
| 32 | 165 | 75 | 124 | 108 | ... |
| 24 | 172 | 81 | 112 | 98 | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
Use summary statistics for quantitative data
Measures of centrality(mean, mode, median)
Percentiles (quartiles, quintiles, deciles)
Measures of spread (range, IQR, variance, standard deviation)
What are the different measures of centrality?
Measures of Centrality
Question: What is the typical value of an attribute in our dataset?
How many runs does Sachin Tendulkar typically score in a match?
How many balls does Sachin Tendulkar typically face in a match?
Measures of Centrality: mean
Notation:
n
data points
x_1, x_2, x_3, ...., x_n
Mean of sample:
\bar{x}
Mean of population:
\mu
x_1 = 0, x_2 = 0, x_3 = 36, x_4 =10, x_5 =20, x_6 = 19
x_{12} = 4, x_{13} = 53, x_{14} = 52, x_{15} = 22 .....
,x_7 = 31, x_8 = 36, x_9 = 53, x_{10} = 30, x_{11} = 0,
...... x_{452} = 52
Measures of Centrality: mean
\bar{x} = \frac{x_1 + x_2 + x_3 + \dots+x_n}{n}
= \frac{1}{n}\sum_{i=1}^{n}x_i
x_1 = 0, x_2 = 0, x_3 = 36, x_4 =10, x_5 =20, x_6 = 19
x_{12} = 4, x_{13} = 53, x_{14} = 52, x_{15} = 22 .....
,x_7 = 31, x_8 = 36, x_9 = 53, x_{10} = 30, x_{11} = 0,
...... x_{452} = 52
Measures of Centrality: mean
\bar{x} = \frac{x_1 + x_2 + x_3 + x_4 + \dots+x_{452}}{452}
= 40.76
x_1 = 0, x_2 = 0, x_3 = 36, x_4 =10, x_5 =20, x_6 = 19
x_{12} = 4, x_{13} = 53, x_{14} = 52, x_{15} = 22 .....
,x_7 = 31, x_8 = 36, x_9 = 53, x_{10} = 30, x_{11} = 0,
...... x_{452} = 52
\bar{x} = \frac{0 + 0 + 36 + 10 + \dots+52}{452}
Measures of Centrality: median
Shikhar Dhawan T20I scores (59 matches)
5, 32, 30, 0, 1, 33, 3, 11, 5, 42, 26, 9, 51, 46, 2, 1, 16, 60, 1, 6, 23, 13, 23, 15, 2, 80, 1, 6, 72, 24, 47, 90, 55, 8, 35, 10, 74, 4, 10, 5, 3, 43, 92, 76, 41, 29, 30, 5, 14, 1, 23, 3, 40, 36, 41, 31, 19, 32, 52
Median is the value which appears at the centre of the data when the data is sorted.
Measures of Centrality: median
Shikhar Dhawan T20I scores (59 sorted scores)
center~location = \frac{n+1}{2}
center~location = \frac{59+1}{2} = 30
\because n = 59~is~odd
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14, 15, 16, 19, 23, 23, 23, 24, 26, 29, 30, 30, 31, 32, 32, 33, 35, 36, 40, 41, 41, 42, 43, 46, 47, 51, 52, 55, 60, 72, 74, 76, 80, 90, 92
Measures of Centrality: median
Shikhar Dhawan T20I scores (59 sorted scores)
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14, 15, 16, 19, 23
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{first 29 elements}}
23, 24, 26, 29, 30, 30, 31, 32, 32, 33, 35, 36, 40, 41, 41, 42, 43, 46, 47, 51, 52, 55, 60, 72, 74, 76, 80, 90, 92
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{last 29 elements}}
There are equal number of elements on either side of the central location
23
\underbrace{~~~~~~~~~~}_{\textit{mid-point}}
30th~element
Measures of Centrality: median
Shikhar Dhawan T20I scores (59 sorted scores)
When n is odd, the median is the value at the central location (or mid-point) which is 23 in this case
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14, 15, 16, 19, 23
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{first 29 elements}}
23, 24, 26, 29, 30, 30, 31, 32, 32, 33, 35, 36, 40, 41, 41, 42, 43, 46, 47, 51, 52, 55, 60, 72, 74, 76, 80, 90, 92
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{last 29 elements}}
23
\underbrace{~~~~~~~~~~}_{\textit{mid-point}}
Measures of Centrality: median
Shikhar Dhawan T20I scores (50 sorted scores)
What happens when n is even? (say is we had data for 50 T20Is only)
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14, 15, 16, 23, 23, 24, 26, 29, 30, 30, 32, 33, 35, 41, 42, 43, 46, 47, 51, 55, 60, 72, 74, 76, 80, 90, 92
Measures of Centrality: median
Shikhar Dhawan T20I scores (50 sorted scores)
\underbrace{~~~~~~~~~~}_{\textit{2 mid-points}}
15
16
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{first 24 elements}}
23, 23, 24, 26, 29, 30, 30, 32, 33, 35, 41, 42, 43, 46, 47, 51, 55, 60, 72, 74, 76, 80, 90, 92
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{last 24 elements}}
There are two mid-points now such that the number of elements of either side is the same
Measures of Centrality: median
Shikhar Dhawan T20I scores (50 sorted scores)
center~locations = \frac{n}{2}~and~\frac{n}{2}+1
\because n = 50~is~even
center~locations = 25~and~26
\underbrace{~~~~~~~~~~}_{\textit{2 mid-points}}
15
16
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{first 24 elements}}
23, 23, 24, 26, 29, 30, 30, 32, 33, 35, 41, 42, 43, 46, 47, 51, 55, 60, 72, 74, 76, 80, 90, 92
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{last 24 elements}}
Measures of Centrality: median
When n is even, the median is the average of the values at the two central locations (or mid-points) which is (15 + 16)/2 = 15.5 in this case
Shikhar Dhawan T20I scores (50 sorted scores)
\underbrace{~~~~~~~~~~}_{\textit{2 mid-points}}
15
16
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{first 24 elements}}
23, 23, 24, 26, 29, 30, 30, 32, 33, 35, 41, 42, 43, 46, 47, 51, 55, 60, 72, 74, 76, 80, 90, 92
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{last 24 elements}}
Measures of Centrality: median
Summary
Data:~x_1, x_2, x_3, \dots, x_n
if~n~is~odd:
median = x_{\frac{n+1}{2}}
if~n~is~even:
median = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2}
the element at position
\frac{n+1}{2}
\large(
\large)
the mean of elements at positions
\frac{n}{2}
\large(
\large)
\&~\frac{n}{2}+1
Measures of Centrality: mode
The mode is defined as the most frequently occurring value in the dataset
Mode = 1
Shikhar Dhawan T20I scores (59 sorted scores)
0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14, 15, 16, 19, 23, 23, 23, 24, 26, 29, 30, 30, 31, 32, 32, 33, 35, 36, 40, 41, 41, 42, 43, 46, 47, 51, 52, 55, 60, 72, 74, 76, 80, 90, 92
Measures of Centrality: mode
Single mode: (only 1 most frequent value)
Multiple modes: (more than 1 most frequent value)
No modes: (all values appear exactly once)
1,2,2,2,3,4,5,5,5,5,5,6,6,7,7,12,12,13,14,15,15,15,15,15,17,18,19,19
1,2,3,7,8,10,13,23,32,43,55,61,65,68,77,85,91,93
the mode is 1
the modes are 5 & 15
5, 32, 30, 0, 1, 33, 3, 11, 5, 42, 26, 9, 51, 46, 2, 1, 16, 60, 1, 6, 23, 13, 23, 15, 2, 80, 1, 6, 72, 24, 47, 90, 55, 8, 35, 10, 74, 4, 10, 5, 3, 43, 92, 76, 41, 29, 30, 5, 14, 1, 23, 3, 40, 36, 41, 31, 19, 32, 52
Measures of Centrality: summary
Summary
Median is the value which appears at the centre of the data when the data is sorted (slight difference when n is odd v/s when n is even)
Mean is the the sum of all the elements in the data divided by the total number of elements
Mode is the most frequent value appearing in the data
What are some characteristics of measures of centrality?
Mean is the centre of gravity
Data:~x_1, x_2, x_3, \dots, x_n
Mean:\bar{x}
The deviation of a point from the mean is defined as the difference between this point and the mean
Deviation:x_i - \bar{x}
Mean is the centre of gravity
The sum of the deviations of all points from the mean is 0
= (x_1 - \bar{x}) + (x_2 - \bar{x}) + \dots + (x_n - \bar{x})
=\sum_{i=1}^{n} (x_i -\bar{x})
(x_1 + x_2 + x_3 + \dots + x_n)
sum of deviations
= \sum_{i=1}^{n} x_i - n\bar{x}
- (\bar{x} + \bar{x} + \dots n~times)
=
=
\sum_{i=1}^{n} x_i - \sum_{i=1}^{n} x_i = 0
(\because \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i)
Mean is the centre of gravity
What is the physical interpretation of the above result?
=0
sum of deviations from the mean
Mean is the centre of gravity
Number line as a seesaw
Imagine
Data points as weights on the seesaw
Weights proportional to deviations from
\bar{x}
Left side values
Right side values
1 2 3 4 5 6 7 8 9 10
Left side values
Right side values
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
Left side values
Right side values
Left side values
Right side values
Mean: 6.0
Left side values
Right side values
1 2 3 4 5 6 7 8 9 10
Mean is the centre of gravity
The mean is thus also called the centre of gravity of the data
Deviations on left side
Deviations on right side
Mean: 6.0
1 2 3 4 5 6 7 8 9 10
=
Sensitivity to outliers
Informally, we define an outlier as any point which is far off from the other values in the data (a formal definition will follow later)
Scores
Frequency
Outlier
Sensitivity to outliers
Alistair Cook: 2, 7, 7, 10, 14, 16, 37, 39, 244
Which player performed better in the series?
Joe Root: 1, 9, 14, 15, 51, 58, 61, 67, 83
Ashes 2017-18 series (runs scored)
Mean = 39.88
Mean = 41.78
Median = 14
Median = 51
Outlier
Sensitivity to outliers
Alistair Cook: 2, 7, 7, 10, 14, 16, 37, 39, 244
Except for the one high score (outlier), Cook performed poorly whereas Root was more consistent (this is reflected in the median but not in the mean)
Joe Root: 1, 9, 14, 15, 51, 58, 61, 67, 83
Mean = 39.88
Mean = 41.78
Median = 14
Median = 51
Outlier
Sensitivity to outliers
Alistair Cook: 2, 7, 7, 10, 14, 16, 37, 39, 244
Old Mean = 41.78
Old Median = 14
Outlier
The mean is very sensitive to outliers whereas the median is not so sensitive
What if we drop the outlier?
New Mean = 16.5
New Median = 12
Sensitivity to outliers (trimmed mean)
Alistair Cook: 2, 7, 7, 10, 14, 16, 37, 39, 244
Mean = 41.78
Trimmed Mean = 18.57
Trimmed mean is computed by dropping k extreme elements from either side (note that we need to drop the same number of elements from both sides)
To account for the sensitivity to outliers it is advised to compute the trimmed mean
Joe Root: 1, 9, 14, 15, 51, 58, 61, 67, 83
Mean = 39.88
Trimmed Mean = 39.28
Sensitivity to outliers (trimmed mean)
Student salaries (INR lakhs) at a top university
Mean = 24.57
Median = 17.5
Trimmed Mean = 18.95
(dropping 2 extreme values on either side)
9.1, 9.4, 10.5, 10.5, 11.5, 11.7, 12.3, 12.7, 12.8, 13.7, 13.8, 14.9, 14.9, 15.3, 16.2, 17.5, 17.6, 18.5, 18.6, 19.3, 19.9, 20.8, 23.6, 23.6, 24.3, 24.4, 32.1, 35.3, 45.5, 98.3, 133.1
Sensitivity to outliers (mode)
The mode is not sensitive to outliers (unless the mode itself is the outlier)
Shikhar Dhawan T20Is scores: 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 6, 8, 9, 10, 10, 11, 13, 14, 15, 16, 19, 23, 23, 23, 24, 26, 29, 30, 30, 31, 32, 32, 33, 35, 36, 40, 41, 41, 42, 43, 46, 47, 51, 52, 55, 60, 72, 74, 76, 80, 90, 92
Mode = 1
Sample: 8, 11, 12, 13, 14, 17, 19, 20, 21, 23, 24, 27, 28, 29, 30, 31, 33, 35, 64, 64
Mode =64
Summary
It is often a good idea to compute a trimmed mean by dropping the same number of elements from both the extremes
Mean is sensitive to outliers whereas median and mode are not
Sensitivity to outliers
What do the measures of centrality look like for different types of distributions?
Perfectly symmetric distribution
If x is the central location in the data then for every element (x-i) in the data, there will also be a corresponding element (x+i)
Can we say something interesting about the mean, median and mode?
Perfectly symmetric distribution
mean = median = mode
mode corresponds to the tallest bar
median also corresponds to the tallest bar with equal no. of elements on either side
What about the mean?
Perfectly symmetric distribution
Toy data: 3,3,3,4,4,4,4,4,5,5,5
What about the mean? (Informal proof)
Let x be the central value (median, 4 here)
Since the data is symmetric for any element x-i (on the left) there will also be an element x+1 (on the right)
\bar{x} = \frac{3+3+3+4+4+4+4+4+5+5+5}{11}
\bar{x} = \frac{(4-1)+(4-1)+(4-1)+4+4+4+4+4+(4+1)+(4+1)+(4+1)}{11}
\bar{x} = \frac{11*4}{11} = 4
Perfectly symmetric distribution
Toy data: 1,2,3,3,3,4,4,4,4,4,5,5,5,6,7
What about the mean? (Informal proof)
Let x be the central value (median, 4 here)
Since the data is symmetric for any element x-i (on the left) there will also be an element x+1 (on the right)
\bar{x} = \frac{1+2+3+3+3+4+4+4+4+4+5+5+5+6+7}{11}
\bar{x} = \frac{(4-3)+(4-2)+(4-1)+(4-1)+(4-1)+4+4+4+4+4+(4+1)+(4+1)+(4+1)+(4+2)+(4+3)}{11}
\bar{x} = \frac{15*4}{15} = 4
Perfectly symmetric distribution
What about the mean? (Informal proof)
The seesaw will be balanced when the fulcrum is placed at the median. Hence
mean = median = mode
Perfectly symmetric distribution
1,2,2,3,3,3,3,4,4,5,6,6,7,7,7,7,8,8,9
mean = median != mode
Mean = 5, Median=5, Mode = 3,7
What about bimodal distributions?
Perfectly symmetric distribution
mean = median
n is even
Other examples of multimodal distributions
Skewed Distributions
Left-skewed distribution: has a long tail to the left (also called negative skewed)
Right-skewed distribution: has a long tail to the right (also called positive skewed)
Left-skewed
Right-skewed
Left Skewed Distributions
[Hint: mean is the centre of the gravity]
Without computing it can you say where would the mean be? (towards left or right)
Mean
Left Skewed Distributions
What about the median and the mode?
Observation: mean < median < mode
Mean
Median
Mode
Left Skewed Distributions
What if the tail is very long?
Observation: mean < median < mode
Mean: 8.91
Median: 9.055
Mode: 9.5
Mean : 8.93
Median: 9.05
Mode: 9.5
Mean: 8.935
Median: 9.06
Mode: 9.5
Right Skewed Distributions
Without computing it can you say where would the mean be? (towards left or right)
[Hint: mean is the centre of the gravity]
Mean
Right Skewed Distributions
What about the median and the mode?
Observation: mean > median > mode
Mean
Median
Mode
Skewed Distributions
Left skewed: mean < median < mode
Right-skewed
Left-skewed
Right skewed: mean > median > mode
Is this always true?
Almost always but not always (counter-example on next slide)
Mean
Mode
Median
Mean
Mode
Median
Skewed Distributions (with heavy and long tail)
Left skewed but mean > median!
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{long tail}}
\underbrace{~~~~~~~~~~~}_{\textit{heavy tail}}
Not true for some skewed distributions which have a heavy tail on the other side
Is this always true?
(not always!)
Median
Mean
mean < median
\underbrace{~~~~~~~~~~~~~~~~~~~~~~~}_{\textit{long tail}}
\underbrace{~~~~~~~~~~~~~~~~}_{\textit{heavy tail}}
(same as the generic rule)
Skewed Distributions (with heavy and long tail)
Mean
Median
Skewed Distributions (bimodal)
Left skewed but mean > median!
Mean
Median
Summary
(Except for some cases where there is a heavy tail on the other side or if the distribution is bimodal)
(Almost always true)
Skewed Distributions
Left skewed: mean < median < mode
Right skewed: mean > median > mode
Left Skewed Distributions
What if the tail is very long?
Observation: mean < median < mode
Mean: 8.91
Median: 9.055
Mode: 9.5
Mean : 8.93
Median: 9.05
Mode: 9.5
Mean: 8.935
Median: 9.06
Mode: 9.5
How do we compute mean, median, mode from histograms?
What if we only have access to histograms and not the actual data?
Histograms (computing measures of centrality)
Can we still compute measures of centrality?
Scores
Frequency
Histograms (computing measures of centrality)
Can we still compute measures of centrality?
| Interval | Frequency |
|---|---|
| 0 - 10 | 126 |
| 10-20 | 59 |
| 20-30 | 44 |
| 30-40 | 47 |
| 40-50 | 31 |
| 50-60 | 22 |
| 60-70 | 28 |
| 70-80 | 10 |
| 80-90 | 18 |
| 90-100 | 18 |
| Interval | Frequency |
|---|---|
| 100-110 | 12 |
| 110-120 | 12 |
| 120-130 | 9 |
| 130-140 | 4 |
| 140-150 | 7 |
| 150-160 | 1 |
| 160-170 | 1 |
| 170-180 | 1 |
| 180-190 | 1 |
| 190-200 | 0 |
| 200-210 | 1 |
Compute median from a histogram
| Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 0 - 10 | 126 | 126 |
| 10-20 | 59 | 185 |
| 20-30 | 44 | 229 |
| 30-40 | 47 | 276 |
| 40-50 | 31 | 307 |
| 50-60 | 22 | 329 |
| 60-70 | 28 | 357 |
| 70-80 | 10 | 367 |
| 80-90 | 18 | 385 |
| 90-100 | 18 | 403 |
| Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 100 - 110 | 12 | 415 |
| 110-120 | 12 | 427 |
| 120-130 | 9 | 436 |
| 130-140 | 4 | 440 |
| 140-150 | 7 | 447 |
| 150-160 | 1 | 448 |
| 160-170 | 1 | 449 |
| 170-180 | 1 | 450 |
| 180-1900 | 1 | 451 |
| 190-200 | 0 | 451 |
| 200-210 | 1 | 452 |
Compute cumulative frequency (find n)
Compute central location
\underbrace{\frac{n+1}{2}}_{n~\textit{is odd}}~~or~~\underbrace{\frac{n}{2}, \frac{n}{2}+1}_{n~\textit{is even}}
Find the interval containing the centre
Estimate median = mid-point of this interval
Compute median from a histogram
| Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 0 - 10 | 126 | 126 |
| 10-20 | 59 | 185 |
| 20-30 | 44 | 229 |
Why does the above procedure make sense?
Median
185 elements
452 -229 = 223 elements
44 elements
0-20
20-30
30-200
Compute median from a histogram
0-20
20-30
30-200
but it is also possible that the 44 elements were different (we don't know what the 44 values are)
20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 28, 28, 29, 29
20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27, 28, 29
185 elements
452 -229 = 223 elements
44 elements
Compute median from a histogram
0-20
20-30
30-200
Since we do not the actual values in the class interval the best guess we can make is that the median is the mid-point of the class interval (we won't be very wrong)
True median: 28
Estimated median: 25
Error:
\frac{28-25}{28}*100=10.71\%
185 elements
452 -229 = 223 elements
44 elements
Compute median from a histogram
Bin Size: 10000
Total yield
# Farms
What if the class intervals are bigger?
Compute median from a histogram
What if the class intervals are bigger?
n =1611
centre = (1611+1)/2 = 806
class interval containing centre =90000-10000
True median: 96080
Estimated median= 95000
Error:
\frac{96080-95000}{96080}*100=1.12\%
The absolute error may be high due to larger values in the data but the relative error may still be reasonable
Compute mean from a histogram
| Interval | Frequency |
|---|---|
| 0 - 10 | 126 |
| 10-20 | 59 |
| 20-30 | 44 |
| 30-40 | 47 |
| 40-50 | 31 |
| 50-60 | 22 |
| 60-70 | 28 |
| 70-80 | 10 |
| 80-90 | 18 |
| 90-100 | 18 |
| Interval | Frequency |
|---|---|
| 100 - 110 | 12 |
| 110-120 | 12 |
| 120-130 | 9 |
| 130-140 | 4 |
| 140-150 | 7 |
| 150-160 | 1 |
| 160-170 | 1 |
| 170-180 | 1 |
| 180-1900 | 1 |
| 190-200 | 0 |
| 200-210 | 1 |
Runs scored
# Matches
How do you compute the mean from a histogram?
Compute mean from a histogram
| Interval | Frequency | Mid-point | Mid-point * frequency |
|---|---|---|---|
| 0 - 10 | 126 | 5 | 630 |
| 10-20 | 59 | 15 | 885 |
| 20-30 | 44 | 25 | 1100 |
| 30-40 | 47 | 35 | 1645 |
| 40-50 | 31 | 45 | 1395 |
| 50-60 | 22 | 55 | 1210 |
| 60-70 | 28 | 65 | 1820 |
| 70-80 | 10 | 75 | 750 |
| 80-90 | 18 | 85 | 1530 |
| 90-100 | 18 | 95 | 1710 |
| 100-110 | 12 | 105 | 1260 |
| 110-120 | 12 | 115 | 1380 |
| 120-130 | 9 | 125 | 1125 |
| 130-140 | 4 | 135 | 540 |
| 140-150 | 7 | 145 | 1015 |
| 150-160 | 1 | 155 | 155 |
| 160-170 | 1 | 165 | 165 |
Compute the mid-point of each interval
Multiply the mid-point by the frequency of the interval
Sum up the resulting product for all intervals
Divide by the no. of data points
| 170-180 | 1 | 175 | 175 |
|---|---|---|---|
| 180-190 | 1 | 185 | 185 |
| 190-200 | 0 | 195 | 0 |
| 200-210 | 1 | 205 | 205 |
sum =18880
mean =
\frac{18880}{452}
=41.769
Compute mean from a histogram
True mean = 40.76
Estimated mean
= 41.77
Error
=\frac{40.76-41.77}{40.76}*100\\=-2.47\%
What is the intuition behind this procedure?
| Interval | Frequency | Mid-point | Mid-point * frequency |
|---|---|---|---|
| 0 - 10 | 126 | 5 | 630 |
| 10-20 | 59 | 15 | 885 |
| 20-30 | 44 | 25 | 1100 |
| 30-40 | 47 | 35 | 1645 |
| 40-50 | 31 | 45 | 1395 |
| 50-60 | 22 | 55 | 1210 |
| 60-70 | 28 | 65 | 1820 |
| 70-80 | 10 | 75 | 750 |
| 80-90 | 18 | 85 | 1530 |
| 90-100 | 18 | 95 | 1710 |
| 100-110 | 12 | 105 | 1260 |
| 110-120 | 12 | 115 | 1380 |
| 120-130 | 9 | 125 | 1125 |
| 130-140 | 4 | 135 | 540 |
| 140-150 | 7 | 145 | 1015 |
| 150-160 | 1 | 155 | 155 |
| 160-170 | 1 | 165 | 165 |
| 170-180 | 1 | 175 | 175 |
|---|---|---|---|
| 180-190 | 1 | 185 | 185 |
| 190-200 | 0 | 195 | 0 |
| 200-210 | 1 | 205 | 205 |
Compute mean from a histogram
What is the intuition behind this procedure?
\bar{x}=\frac{\sum_{i=1}^{n} x_i}{n}
\sum_{i=1}^{n} x_i =
sum of elements in 1st interval
+ sum of elements in 2nd interval
+ sum of elements in 3rd interval
+ sum of elements in last interval
... ... ... ...
| Interval | Frequency |
|---|---|
| 0 - 10 | 126 |
| 10-20 | 59 |
| 20-30 | 44 |
| 30-40 | 47 |
| 40-50 | 31 |
| 50-60 | 22 |
| 60-70 | 28 |
| 70-80 | 10 |
| 80-90 | 18 |
| 90-100 | 18 |
| .... ..... | .... .... |
Compute mean from a histogram
Problem: We do not know what these 10 values are?
sum of elements in 8th interval
= sum of 10 elements
| Interval | Frequency |
|---|---|
| 0 - 10 | 126 |
| 10-20 | 59 |
| 20-30 | 44 |
| 30-40 | 47 |
| 40-50 | 31 |
| 50-60 | 22 |
| 60-70 | 28 |
| 70-80 | 10 |
| 80-90 | 18 |
| 90-100 | 18 |
| .... ..... | .... .... |
Solution: Assume each value is equal to the mid-point (over-estimate some values, underestimate some values)
70, 70, 71, 72, 73, 74, 77, 77, 78, 79
mid-point = 75, True sum = 741
Approximation: add 75 10 times = 75 * 10
What if the class intervals are bigger?
True mean: 135812.14
Estimated mean: 136837.37
Error:
\frac{135812.14-136837.37}{135812.14}*100=0.75\%
The absolute error may be high due to larger values in the data but the relative error may still be reasonable
Compute mean from a histogram
It is not possible to compute the mode from a histogram if the bin size is greater than 1
Compute mode from a histogram
2, 2, 11, 13, 14, 15, 16, 19, 20, 22, 23, 24, 26, 29, 31, 32, 35, 36, 37, 38, 39, 41, 42, 43, 44, 48, 49, 50, 52, 53, 54, 55, 56, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69
Compute mode from a histogram
Of course if the bin size is 1 it is trivial to compute the mode
Modes: 0 and 1
What is the effect of transformations on the measures of centrality?
Transformations
Scaling: (example, kilometres to metres)
x_{new} = a*x
(a = 1000)
(a = 0.4535)
Original Data: (Distance km)
[50.52, 62.935, 50.888, 62.94, 62.929, 37.8, 36.687, 39.122, 63.453, 44.845]
Scaled Data: (Distance m)
[50520.0, 62935.0, 50888.0, 62940.0, 62929.0, 37800.0, 36687.0, 39122.0, 63453.0, 44845.0]
Original Data:(weight lbs)
[13, 29, 21, 34, 30, 33, 11, 31, 15, 20]
Scaled Data: (weight kgs)
[5.89, 13.15, 9.52, 15.41, 13.60, 14.96, 4.98, 14.05, 6.80, 9.07]
Transformations
Shifting: (flat 50 INR off on shirts)
x_{new} = x + c
(c = -50)
(c = 5)
Shifting: (flat 5 INR packing charge per item)
Original Data: (Pre discount)
[699, 599, 549, 1499, 799, 999, 1150, 850, 899, 1099]
Shifted Data: (Post Discount)
[649, 549, 499, 1449, 749, 949, 1100, 800, 849, 1049]
{'Veg Burger': 35, 'Cheese Maggi':45, 'Masala Dosa':80, 'Fried Rice': 75, 'Pizza': 129}
Original Data:
{'Veg Burger': 40, 'Cheese Maggi':50, 'Masala Dosa':85, 'Fried Rice': 80, 'Pizza': 134}
Shifted Data: (Post Packing)
Transformations
Scaling and Shifting
x_{new} = a*x + c
(a = 5/9, c = -160/9)
Temperature in Fahrenheit:
[75.25, 71 , 55.15, 58.28, 69.71, 44.4 , 38.77, 44.96, 80.7 , 73.76]
Temperature in Celsius:
[24.03, 21.67, 12.86, 14.6 , 20.95, 6.89, 3.76, 7.2 , 27.06, 23.2]
Transformations
Summary
Scaling and Shifting:
x_{new} = a*x + c
Special cases:
c=0: x_{new} = a*x
(Only scaling)
(Only shifting)
a=1: x_{new} = x+c
Effect of transformations on mean
Prove that if
x_{new} = a*x + c
then,
\bar{x}_{new} = a*\bar{x} + c
Proof:
\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
\bar{x}^{new} = \frac{1}{n}\sum_{i=1}^{n} x^{new}_i
= a\bar{x} + c
= \frac{1}{n}\sum_{i=1}^{n} (\overbrace{a x_i + c})
= \frac{1}{n}(\sum_{i=1}^{n} a x_i + \sum_{i=1}^{n} c )
= a*\frac{1}{n}\sum_{i=1}^{n} x_i + \frac{1}{n} * nc
Effect of transformations on mean
\bar{x} = 61.2
\bar{x}^{new} = a*\bar{x} + c
(a = 5/9, c = -160/9)
\bar{x}^{new} = \frac{5}{9}*\bar{x} + \frac{-160}{9} = 16.2
Temperature in Celsius:
[24.03, 21.67, 12.86, 14.6 , 20.95, 6.89, 3.76, 7.2 , 27.06, 23.2]
Temperature in Fahrenheit:
[75.25, 71 , 55.15, 58.28, 69.71, 44.4 , 38.77, 44.96, 80.7 , 73.76]
Effect of transformations (on median)
Temperature in Fahrenheit:
Temperature in Celsius:
x_{new} = a*x + c
median_{new} = a*median + c
[3.76, 6.89, 7.2 , 12.86, 14.6 , 20.95, 21.67]
[38.77, 44.4 , 44.96, 55.15, 58.28, 69.71, 71]
The location of the median does not change (it only gets scaled)
12.86 = \frac{5}{9}*55.15 + \frac{-160}{9}
Effect of transformations (on mode)
Temperature in Fahrenheit:
Temperature in Celsius:
The scaled value of the mode will be the new mode
x_{new} = a*x + c
mode_{new} = a*mode + c
[3.76, 6.91, 6.91, 12.86, 14.6 , 20.95, 21.67, 23.2 , 24.03, 27.06]
[38.77, 44.44 , 44.44, 55.15, 58.28, 69.71, 71. , 73.76, 75.25, 80.7]
6.91 = \frac{5}{9}*44.44 + \frac{-160}{9}
Summary
mean
Mean is sensitive to outliers but median is not
median
mode
\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
x_{\frac{n+1}{2}}~or~ \frac{x_{\frac{n}{2}} + x_{\frac{n}{2} + 1}}{2}
most freq. element
Mean is the centre of gravity of the data
Summary
Left skewed: mean < median < mode
Right skewed: mean > median > mode
Symmetric: mean = median = mode
Effect of Skewness
Almost always
\large(
\large)
Mean and median can be approximately computed from histograms
Effect of Transformations:
x_{new} = a*x + c
\bar{x}_{new} = a*\bar{x} + c
median_{new} = a*median + c
mode_{new} = a*mode + c
Left-skewed-histogram: Most of the short bars are towards the left of the histogram
Typical trends in histograms
Units
Frequency
Average Strike Rate
Frequency
Units
Frequency
Measures of Centrality
| Age | Height | Weight | Cholesterol | Sugar level | .... ..... |
|---|---|---|---|---|---|
| 32 | 165 | 75 | 124 | 108 | ... |
| 24 | 172 | 81 | 112 | 98 | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
Which is the 7th most grown crop in the country?
Hard to answer
Nominal attributes
Recall
Question: What is the typical value of an attribute in our dataset?
How many runs does Sachin Tendulkar typically score in a match?
How many balls does Sachin Tendulkar typically face in a match?
Measures of Centrality
| Age | Height | Weight | Cholesterol | Sugar level | .... ..... |
|---|---|---|---|---|---|
| 32 | 165 | 75 | 124 | 108 | ... |
| 24 | 172 | 81 | 112 | 98 | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
Which is the 7th most grown crop in the country?
Hard to answer
Nominal attributes
Recall
Question: What is the typical value of an attribute in our dataset?
How many runs does Sachin Tendulkar typically score in a match?
How many balls does Sachin Tendulkar typically face in a match?
Motivation: Summarise Big Data
| Age | Height | Weight | Cholesterol | Sugar level | .... ..... |
|---|---|---|---|---|---|
| 32 | 165 | 75 | 124 | 108 | ... |
| 24 | 172 | 81 | 112 | 98 | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
| ... | ... | ... | ... | ... | ... |
Which is the 7th most grown crop in the country?
Hard to answer
Nominal attributes
| Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 0 - 10 | 137 | 196 |
| 10-20 | 59 | 196 |
| 20-30 | 44 | 240 |
| Interval | Frequency | Mid-point | Mid-point * frequency |
|---|---|---|---|
| 0 - 10 | 137 | 5 | 685 |
| 10-20 | 59 | 15 | 885 |
| 20-30 | 44 | 25 | 1100 |
| 30-40 | 47 | 35 | 1645 |
| 40-50 | 31 | 45 | 1395 |
| 50-60 | 22 | 55 | 1210 |
| 60-70 | 28 | 65 | 750 |
| 70-80 | 10 | 75 | 1530 |
| 80-90 | 18 | 85 | 1710 |
| 90-100 | 18 | 95 | 1260 |
| 100-110 | 12 | 105 | 1260 |
| 110-120 | 12 | 115 | 1380 |
| 120-130 | 9 | 125 | 1125 |
| 130-140 | 4 | 135 | 540 |
| 140-150 | 7 | 145 | 1015 |
| 150-160 | 1 | 155 | 155 |
| 160-170 | 1 | 165 | 165 |
| 170-180 | 1 | 175 | 175 |
|---|---|---|---|
| 180-190 | 1 | 185 | 185 |
| 190-200 | 0 | 195 | 0 |
| 200-210 | 47 | 205 | 205 |
Left side values
Right side values
Mean: 6.0
Left side values
Right side values
1 2 3 4 5 6 7 8 9 10
Left side values
Right side values
1 2 3 4 5 6 7 8 9 10
Left side values
Right side values
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10