Skewness in distributionsWhen a distribution is symmetric, the mean and the mode coincide. If the distribution is left-skewed or negative, the mean is less than the mode; and if it is right-skewed or positive, the mean is greater than the mode.Thus:\[ \bar{x} - Mo = 0 \text{--> null skewness or symmetric distribution} \]\[ \bar{x} - Mo < 0 \text{--> negative or left skewness} \]\[ \bar{x} - Mo > 0 \text{--> positive or right skewness} \]Types of skewnessWhich frequencies below are symmetric?classesFrequency2-666-101210-142414-181218-226We have:\[ \bar{x} = 12 \]\[ Md = 12 \]\[ Mo = 12 \]\[ s = 4,42 \]Therefore, the distribution is symmetric.classesFrequency2-666-101210-142414-183018-226We have:\[ \bar{x} = 12,9 \]\[ Md = 13,5 \]\[ Mo = 16 \]\[ s = 4,20 \]Therefore, the distribution is negatively skewed.classesFrequency2-666-103010-142414-181218-226We have:\[ \bar{x} = 11,1 \]\[ Md = 10,5 \]\[ Mo = 8 \]\[ s = 4,20 \]Therefore, the distribution is positively skewed.Coefficient of skewnessThe previous measure of skewness, being absolute, has the same limitation as the standard deviation; that is, it does not allow comparisons with other distributions in order to discuss whether one is more skewed than another.For comparisons, the Pearson coefficient of skewness is used:\[ As = \frac{3 · (\bar{x} - Md)}{s} \]From the previous examples:\[ As_1 = \frac{3 · (12-12)}{4,42} = 0 \]\[ As_2 = \frac{3 · (12,9-13,5)}{4,20} = -0,429 \]\[ As_3 = \frac{3 · (11,1 - 10,5)}{4,20} = 0,429 \]Kurtosis> We call kurtosis the degree of flattening in relation to a standard normal distribution.Kurtosis indicates how concentrated the data are around the mean. It tells us whether the curve is more "pointed" or "flattened" compared to the normal distribution.Type of kurtosisTechnical nameCharacteristicsHighLeptokurticHigh peak and long tails – more data concentrated at the mean, but many extreme values.NormalMesokurticShape similar to the normal curve.LowPlatykurticFlattened curve – more dispersed data, few extremes.Examples:1. A distribution of grades with many students with an average close to 7, but also many with a grade of 0 or 10 → Leptokurtic2. A height distribution with little variation among individuals → Mesokurtic3. An age distribution where all age groups appear with similar frequency → PlatykurticKurtosis Coefficient (excess kurtosis)Kurtosis can be quantified as follows:\[ K = \frac{ \sum (x_i - \bar{x})^4 }{n s^4} \]The standard measure of the kurtosis of a distribution, originated by Karl Pearson, is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not to its peak; therefore, the sometimes-seen characterization of kurtosis as "peak" is incorrect. For this measure, higher kurtosis corresponds to the greater extremity of deviations (or outliers), and not to the configuration of the data near the mean.But we more often use excess kurtosis, defined as:\[ \text{Excess kurtosis} = K - 3 \]- If \( K - 3 > 0 \): Leptokurtic- If \( K - 3 = 0 \): Mesokurtic- If \( K - 3 < 0 \): PlatykurticThe value 3 represents the kurtosis of the Normal distribution, so this is the value used as the standard for comparison.Boxplot or Box-and-Whisker Plot to check Skewness and KurtosisDeveloped by John Tukey in 1970, the boxplot is a visual representation that summarizes the distribution of a data set through five summary measures: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Its graphical simplicity allows a quick and efficient analysis of the dispersion, skewness, and outliers of the data.Boxplot and its points explainedElements of the Boxplot:- Box: Represents the interquartile range (IQR = Q3 - Q1), containing 50% of the central data- Median line: Divides the box into two parts, indicating the center of the distribution- Whiskers: Extend up to 1.5 × IQR below Q1 and above Q3- Outliers: Points beyond the whiskers, indicating extreme observationsThe boxplot is an exceptional tool for visually evaluating the skewness of a distribution because its structure immediately reveals the relative positioning of the median (the line inside the box) and the symmetry of the quartiles. In a symmetric distribution, the median will be approximately in the center of the box, and the lengths of the upper and lower "whiskers" will be similar. Skewness, on the other hand, is identified intuitively: if the median is closer to the lower part of the box (Q1) and the upper whisker is considerably longer than the lower one, the distribution is skewed to the right (positive). Conversely, if the median is close to the top of the box (Q3) and the lower whisker is longer, the skewness is to the left (negative). This visual representation of the quartiles and the extent of the tails provides an immediate and robust understanding of the symmetry of the data, without the need for complex calculations.Comparisons between distributions using boxplotIn addition to skewness, the boxplot offers valuable insights into kurtosis, which relates to the "flattening" of the distribution and the weight of the tails. A compact box (with a small Interquartile Range - IQR) combined with very long whiskers indicates a leptokurtic distribution, where the data are highly concentrated around the center but have heavy and long tails, with the presence of extreme values. On the other hand, a wide box (large IQR) and short whiskers suggest a platykurtic distribution, where the data are more spread out and there are no values extremely far from the center. In this way, the combination of the ratio between the width of the box and the length of the whiskers, together with the detection of outliers, allows a clear inference about kurtosis and the general nature of the distribution, making the boxplot a powerful tool for exploratory data analysis.Building the BoxplotLet's use the following numbers, already ordered:1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39Put your data in orderThe data are already in ascending order.Find the Median (Q2)Since we have 40 numbers (n = 40, even), the median will be the average of the two central values:- Central positions: 20th and 21st- Values: 19 and 20- Median (Q2) = (19 + 20) / 2 = 19.5Find the First Quartile (Q1)Q1 is the median of the first half of the data (first 20 numbers):- First half: 1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19- Median of the first 20: average of the 10th and 11th values- Values: 9 and 10- Q1 = (9 + 10) / 2 = 9.5Find the Third Quartile (Q3)Q3 is the median of the second half of the data (last 20 numbers):- Second half: 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39- Median of the last 20: average of the 10th and 11th values of this half- Values: 29 and 30- Q3 = (29 + 30) / 2 = 29.5Calculate the Interquartile Range (IQR)IQR = Q3 - Q1 = 29.5 - 9.5 = 20Calculate the Limits to Identify Outliers- Lower Limit = Q1 - 1.5 × IQR = 9.5 - 1.5 × 20 = 9.5 - 30 = -20.5- Upper Limit = Q3 + 1.5 × IQR = 29.5 + 1.5 × 20 = 29.5 + 30 = 59.5Identify OutliersCheck whether any value is below -20.5 or above 59.5:- Minimum value in the data: 1- Maximum value in the data: 39- There are no outliers in this set, since all values are within the interval [-20.5, 59.5].Find the Whiskers- Lower whisker: smallest value within the lower limit → 1- Upper whisker: largest value within the upper limit → 39Build the BoxplotNow we have all the measures:- Minimum: 1- Q1: 9.5- Median (Q2): 19.5- Q3: 29.5- Maximum: 39Schematic drawing:0 5 10 15 20 25 30 35 40|----|----|----|----|----|----|----|----||---------|__________|---------|Q1(9.5) Med(19.5) Q3(29.5)Whisker: from 1 to 9.5 and from 29.5 to 39Exercises1) Consider the following results related to three frequency distributions:DistributionsmeanModeA5252B4550C4846Determine the type of skewness of each one.2) A frequency distribution has the following measures: mean = 48.1; median = 47.9; and standard deviation = 2.12. Calculate Pearson’s skewness coefficient.3) In a frequency distribution, the following measures were found:Mean = 33.18; mode = 27.50; median = 31.67; and standard deviation = 12.25.a. Classify the type of skewness.b. Calculate Pearson’s skewness coefficient.4. A distribution has Kurtosis = 4.5. Classify the distribution (platykurtic, mesokurtic, or leptokurtic).5. Explain in your own words what a leptokurtic distribution with negative skewness means.6. Create the Boxplot for the following set: [12, 14, 19, 22, 23, 25, 27, 29, 31, 34, 35, 37, 39, 42, 44, 45, 48, 51, 53, 56, 59, 61, 63, 66, 67, 68, 71, 74, 77, 78, 79, 82, 85, 86, 88, 91, 93, 95, 97]7. Create the Boxplot for the following set: [-15, 5, 29, 31, 32, 33, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 88, 120, 150]
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