Data Distribution

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Measures of Central Tendency:
A single number that describes the central, typical, or representative value of a dataset, e.g, mean, median, and mode.
The mean is the average, the median is the middle value in a sorted list, and the mode is the most frequently occurring value.

• A single representative value can be used to compare different groups or distributions.

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Mean:
The artihmetic average of a set of numbers i.e sum all values and divide by the number of values.
\(mean = \frac{1}{n}\sum_{i=1}^{n}x_i\)

• Most common measure of central tendency.
• Represents the ‘balancing point’ of data.
• Sample mean is denoted by \(\bar{x}\), and population mean by \(\mu\).

Pros:

• Uses all datapoints in its calculation, providing a comprehensive measure.

Cons:

• Highly sensitive to outliers i.e exteme values.

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Median:
The middle value of a sorted list of numbers. It divides the dataset into 2 equal halves.
Calculation:

• Arrange the data points in ascending order.
• If the number of data points is even, the median is the average of the two middle values.
• If the number of data points is odd, the median is the middle value i.e \((\frac{n+1}{2})^{th}\) element.

Pros:

• Not impacted by outliers, making it a more robust/reliable measure, especially for skewed distributions.

Cons:

• Does NOT use all the datapoints in its calculation.

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Mode:
The most frequently occurring value in a dataset.

• Dataset can have 1 mode i.e unimodal, 2 modes i.e bimodal, and more than 2 modes i.e multimodal.
  • If NO value repeats, then NO mode.

Pros:

• Only measure of central tendency that can be used for categorical/nominal data, such as, gender, blood group, level of education, etc.
• It can reveal important peaks in data distribution.

Cons:

• A dataset can have multiple modes, or no mode at all, which can make mode less informative.

📘 Measures of Dispersion(Spread):
It measures the spread or variability of a dataset.
Quantifies how spread out or scattered the data points are.
E.g: Range, Variance, Standard Deviation, Median Absoulute Deviation(MAD), Skewness, Kurtosis, etc.

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Range:
The difference between the largest and smallest values in a dataset. Simplest measure of dispersion
\(range = max - min\)

Pros:

• Easy to calculate and understand.

Cons:

• Only considers the the 2 extreme values of dataset and ignores the distribution of data in between.
• Highly sensitive to outliers.

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Variance:
The average of the squared distance of each value from the mean.
Measures the spread of data points.

\(sample ~ variance = s^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2\)

\(population ~ variance = \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2\)

Cons:

• Highly sensitive to outliers, as squaring amplifies the weight of extreme data points.
• Less intuitive to understand, as the units are square of original units.

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Standard Deviation:
The square root of the variance, measures average distance of data points from the mean.

• Low standard deviation indicates that the data points are clustered around the mean, whereas
  high standard deviation means that the data points are spread out over a wide range.

\(s = sample ~ standard ~ deviation \)
\(\sigma = population ~ standard ~ deviation \)

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Mean Absolute Deviation:
It is the average of absolute deviation or distance of all data points from mean.

\( mad = \frac{1}{n}\sum_{i=1}^{n}|x_i - \bar{x}| \)

Pros:

• Less sensitive to outliers as compared to standard deviation..
• More intuitive and simpler to understand.

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Skewness:
It measures the asymmetry of a data distribution.
Tells us whether the data is concentrated on one side of mean and is there a long tail stretching on the other side.

Positive Skew:

• Tail is longer on the right side of the mean.
• Bulk of data is on the left side of the mean, but there are a few very high values pulling the mean towards the right.
• Mean > Median > Mode.

Negative Skew:

• Tail is longer on the left side of the mean.
• Bulk of data is on the right side of the mean, but there are a few very high values pulling the mean towards the left.
• Mean < Median < Mode.

Zero Skew:

• Perfectly symmetrical like a normal distribution.
• Mean = Median = Mode.

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Kurtosis:
It measures the “tailedness” of a data distribution.
It describes how much the data is concentrated in tails (fat or thin) versus the center.

• It can tell us about the frequency of outliers in the data.
  • Thick tails => More outliers.

Excess Kurtosis:
Excess kurtosis is calculated by subtracting 3 from standard kurtosis in order to compare with normal distribution.
Normal distribution has kurtosis = 3.

Mesokurtic:

• Excess kurtosis = 0 i.e normal kurtosis.
• Tails are neither too thick nor too thin.

Leptokurtic:

• High kurtosis, i.e, excess kurtosis > 0 (+ve).
• Heavy or thick tails => High probability of outliers.
• Sharp peak => High concentration of data around mean.
• E.g: Student’s t-distribution, Laplace distribution, etc.
• High risk stock portfolios.

Platykurtic:

• Low kurtosis, i.e, excess kurtosis < 0 (-ve).
• Thin tails => Low probability of outliers.
• Low peak => more uniform distribution of values.
• E.g: Uniform distribution, Bernoulli(P=0.5) distribution, etc.
• Investment in fixed deposits.

📘 Measures of Position:
It helps us understand the relative position of a data point i.e where a specific value lies within a dataset.
E.g: Percentile, Quartile, Inter Quartile Range(IQR), etc.

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Percentile:
It indicates the percentage of scores in a dataset that are equal to or below a specific value.
Here, the complete dataset is divided into 100 equal parts.

• \(k^{th}\) percentile => at least \(k\) percent of the data points are equal to or below the value.
• It is a relative comparison, i.e, compares a score with the entire group’s performance.
• Quartiles are basis for box plots.

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Quartile:
They are special percentiles that divide the complete dataset into 4 equal parts.

Q1 => 25th percentile, value below which 25% of the data falls.
Q2 => 50th percentile, value below which 50% of the data falls; median.
Q3 => 75th percentile, value below which 75% of the data falls.

\[ Q1 = (n+1) * 1/4 \\ Q2 = (n+1) * 1/2 \\ Q3 = (n+1) * 3/4 \]

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Inter Quartile Range(IQR):
It is the single number that measures the spread of middle 50% of the data, i.e Q1-Q3.

• More robust measure of spread than range as is NOT impacted by outliers.

IQR = Q3 - Q1