Statistical and graphical analysis: univariate analysis

Statistical and graphical analysis: univariate analysis

When we talk about univariate analysis, we will normally find elements such as:

  • Histograms.
  • Percentiles: The percentile (or centile) is the value of the variable below which a certain percentage of observations fall; for example, the 20th percentile is the value (or score) below which 20 per cent of the observations fall.
  • Statistical moments: Mean, median, standard deviation.
  • The moment above zero: Mean, median, second moment.
  • Moments above the mean: Variance, standard deviation, skewness, kurtosis.
Kurtosis formula and skewness formula
Kurtosis formula and skewness formula

In the statistical and graphical analysis, specifically in the univariate analysis, we can observe two main elements to take into account:

  • Kurtosis: The fourth central moment is whether the distribution is tall and thin or short and square, compared to the normal distribution of the same variance.
  • Skewness: The measure of the skewness of the probability distribution of a real-valued random variable. The value of skewness can be positive or negative or even undefined. Qualitatively, a negative skewness value indicates that the tail of the left-hand side of the probability density function is longer than that of the right-hand side. Most of the values (possibly including the median) lie to the right of the mean.

Kurtosis

From the highest peak to the lowest peak:

Kurtosis
types of kurtosis

Kurtosis: Leptokurtic, Mesokurtic and Platykurtic:

  • Leptokurtic: “Lepto” (meaning thin). In terms of shape, a leptokurtic distribution has a sharper peak around the mean.
  • Mesocurtic: Semicircular distribution. Elevation in the cosine distribution. Uniform distribution.
  • Platicurtic: “Plati” comes from tabla. It has a platicurtic distribution with a lower and broader peak around the mean in terms of shape.

Skewness

types of asymmetry: negative and positive skewness
types of asymmetry: negative and positive skewness

Types of asymmetry: negative and positive skewness

  • Negative skewness: The height on the left is longer; the mass of the distribution is concentrated on the figure’s right. The distribution is said to be left (observations): 1,1001,1002,1003.
  • Positive skewness: The high on the right is longer; the mass of the distribution is concentrated on the figure’s left. It has relatively few high values. The distribution is said to be right (observation) 1,2,3,1000.

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