## **Skewed Distribution ****| skewed distribution definition**

**Skewed Distribution**

**| skewed distribution definition**The distribution in which symmetry is not present and the curve of the distribution shows tail either left or right side is known as skewed distribution, so skewness is the asymmetry present in the curve or histogram apart from the symmetric or normal curve.

depending on the measure of central tendencies the nature of the distribution whether skewed or not can be evaluated there is special relations between mean, mode and median in left-tailed or right-tailed skewed distribution.

**normal distribution vs skewed | normal vs skewed distribution**

Normal distribution | skewed distribution |

In Normal distribution the curve is symmetric | In skewed distribution the curve is not symmetric |

The measure of central tendencies mean, mode and median are equal | The measure of central tendencies mean, mode and median are not equal |

mean=median =mode | mean>median>mode or mean<median<mode |

**skewed distribution examples in real life**

skewed distribution occurs in number of real life situation like the ticket sale of the particular show or movies in different months, record of athletes performance in competition, stock market returns, real estate rates fluctuation, life cycle of specific species, income variation, exam score and many more competitive outcomes. The distribution curve which shows asymmetry occurs frequently in applications.

## **difference between symmetrical and skewed distribution | symmetrical and skewed distribution**

The main difference between the symmetrical distributions and skewed distribution is the differences between the central tendencies mean median and mode and in addition as the name suggest in the symmetrical distribution the curve of distribution is symmetric while in the skewed distribution the curve is not symmetric but have the skewness and it may be right-tailed or left tailed or may be both tailed also, the different distribution differs only on the nature of the skewness and symmetry so all the probability distributions can be classified into these two main categories.

To find the nature of distribution whether symmetric or skewed we must have to either draw the curve of the distribution or the coefficient of skewness with the help of absolute or relative measures.

## **highly skewed distribution**

The modal or highest value of the distribution if differs from mean and median that gives the skewed distribution, if the highest value coincides with mean and median and equal then the distribution is symmetric distribution, the highly skewed distribution may be positive or negative. The skewed distribution modal value can be find out using the coefficient of skewness.

**Negatively skewed distribution| which is a negatively skewed distribution**

Any distribution in which the measure of central tendencies follows the order **mean<median<mode **and the coefficient of skewness in negative in the negatively skewed distribution, the negatively skewed distribution is also known as left skewed distribution because in negatively skewed distribution the tail of graph or plot of information is left.

The coefficient of skewness for the negatively skewed distribution can easily find out with the usual methods of finding the coefficients of skewness.

## negatively skewed distribution example

If 150 students in an examination performed as given below then find the nature of skewness of the distribution

marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

freq | 12 | 40 | 18 | 0 | 12 | 42 | 14 | 12 |

**Solution:** To find the nature of skewness of distribution we have to calculate the coefficient of skewness for which we require mean, mode, median and standard deviation for the given information so for this we will calculate these with the help of the following table

class interval | f | mid value x | c.f. | d’=(x-35)/10 | f*d’ | f*d’^{2} |

0-10 | 12 | 5 | 12 | -3 | -36 | 108 |

10-20 | 40 | 15 | 52 | -2 | -80 | 160 |

20-30 | 18 | 25 | 70 | -1 | -18 | 18 |

30-40 | 0 | 35 | 70 | 0 | 0 | 0 |

40-50 | 12 | 45 | 82 | 1 | 12 | 12 |

50-60 | 42 | 55 | 124 | 2 | 84 | 168 |

60-70 | 14 | 65 | 138 | 3 | 42 | 126 |

70-80 | 12 | 75 | 150 | 4 | 48 | 192 |

total=52 | total=784 |

so the measures will be

[latex]\begin{array}{l}

Median =\mathrm{L}+\frac{\left(\frac{\mathrm{N}}{2}-\mathrm{C}\right)}{\mathrm{f}} \times \mathrm{h}=40+\frac{75-70}{10} \times 10=45

\\Mean (\overline{\mathrm{x}})=\mathrm{A}+\frac{\sum_{\mathrm{i}=1}^{\mathrm{k}} \mathrm{fd}^{\prime}}{\mathrm{N}} \times \mathrm{h}=35+\frac{52}{150} \times 10=39.16

\end{array}[/latex]

and

[latex]\begin{aligned}

Standard Deviation }(\sigma) &=\mathrm{h} \times \sqrt{\frac{\sum \mathrm{fd}^{\prime 2}}{\mathrm{~N}}-\left(\frac{\sum \mathrm{fd}}{\mathrm{N}}\right)^{2}} \\ &=10 \times \sqrt{\frac{784}{150}-\left(\frac{52}{150}\right)^{2}}

\\&=10 \times \sqrt{5.10}=22.38 \end{aligned}[/latex]

hence the coefficient of skewness for the distribution is

[latex]S_k=\frac{3(Mean-Median)}{\sigma}

\\=\frac{3(39.16-45)}{22.38}=-0.782[/latex]

## negatively skewed distribution mean median mode

In the negatively skewed distribution mean median mode is in ascending order which represents the tail on the left side of the curve of distribution, the measure of central tendencies mean median and mode for the negatively skewed distribution follows exactly the reverse pattern of positively skewed distribution. The curve of the negatively skewed distribution is also an inverse image of the positively skewed distribution. so Mean<median<mode in negatively skewed distribution.

## negatively skewed distribution curve

The nature of the curve for the negatively skewed distribution curve is left-skewed without symmetry either in a histogram or continuous curve.

As symmetry is the measure to calculate the asymmetry present in the distribution, so the distribution curve of negatively skewed distribution shows the asymmetry present on the left side.

## positively skewed normal distribution

The continuous distribution which is following the normal distribution curve including the asymmetry by gathering the information to the right tail shows the right-skewed curve asymmetric about the median following descending order in the central tendencies mean median and mode.

## FAQs

## Why chi square distribution is positively skewed

The chi-square distribution gives the values from zero to infinity and the curve of the distribution gathers the information in the right tail so it shows the right-skewed curve hence the chi-square distribution is a positively skewed distribution.

## Is Poisson distribution positively skewed

Yes, Poisson distribution is a positively skewed distribution as the information scattered near the right tail so the nature of the plot is positively skewed

## Why does negative binomial distribution always positively skew

The negative binomial distribution is always positively skewed because negative binomial distribution is the generalization of pascal distribution which is always positively skewed so is the negative binomial distribution.

## Does skewness have any impact on linear regression models My dependent variable and my interaction variable is positively skewed

The impact on linear regression of the model having my dependent variable and my interaction skewed does not mean the regression error is also skewed and vice versa as the error is skewed does not mean the variables are skewed.