قياس انحراف التوزيع

Measuring Distribution Skewness

• Determining distribution symmetry from frequency graphs (histograms or curves)
• Skewness identifies deviation from symmetry
• Right-skewed: peak on left, positive skew, mean > median > mode
• Left-skewed: peak on right, negative skew, mean < median < mode
• Symmetrical: no skew, mean = median = mode
• Skewness degree: measures how far data points deviate from the symmetrical distribution
• Skewness is identified by the shape of the tail of a distribution
• Right-skewed: right tail longer
• Left-skewed: left tail longer
• Magnitude of skewness: Determines the central tendency representative measure
• Low skewness indicates symmetry, mean is a good representative measure
• High skewness indicates asymmetry, median is a better measure

Skewness Coefficient

• Measures the degree of non-symmetry in a distribution
• Positive coefficient: right-skewed
• Negative coefficient: left-skewed
• Zero coefficient: symmetrical distribution

Pearson's First Skewness Coefficient

• Measures skewness using mean and mode
• SK = (Mean - Mode) / Standard Deviation

Pearson's Second Skewness Coefficient

• Measures skewness using mean and median
• SK = 3(Mean - Median) / Standard Deviation

Identifying Outliers

• Outliers are extreme values that differ significantly from other data points.
• Method to identify outliers:
• Calculate inner fences: Lower fence (LF) = Q1 - 1.5IQR and Upper fence (UF) = Q3 + 1.5IQR
• Any value below LF or above UF is considered an outlier.
• IQR is the interquartile range (Q3 - Q1)

Box-and-Whisker Plot

• Visual representation displaying data distribution
• Based on five summary measures:
• Minimum value
• First quartile (Q1)
• Median
• Third quartile (Q3)
• Maximum value
• The box represents the IQR, whiskers extend to the non-outlier data points
• Outliers are plotted individually as asterisks (*).
• Skewness: if median is not in the middle of the box= skew
• Symmetry: median is clearly in the middle of the box.

Standardized Values (Z-scores)

• Express data point relative to the mean and standard deviation
• Calculation : Z = (x - mean)/standard Deviation
• Standardized values (z-scores) measure how many standard deviations a data point is from the mean
• Positive z-scores: above the mean
• Negative z-scores: below the mean
• Z-score = 0: exactly at the mean
• Use z-scores for comparing values in different distributions (comparing with another set of data)

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