# How to use the SKEW function

**What is the SKEW function?**

The SKEW function calculates the skewness of a group of values with an asymmetric tail from its mean value.

## 1. Introduction

**What is skewness?**

Skewness and asymmetric tails describe the lack of balance and symmetry in probability distributions and datasets. Identifying skewness is fundamental in statistics.

Skewness provides crucial information about the shape and symmetry of a distribution. A skewed distribution deviates from a perfectly symmetrical bell curve, indicating that the data is not evenly distributed around the mean or median.

Many statistical tests and analyses assume that the data follows a normal (Gaussian) distribution. Skewness is a key indicator of non-normality.

Skewness can be an indicator of outliers or extreme values in the data. Distributions with high skewness may be heavily influenced by a few extreme observations which can impact statistical analyses.

Skewness influences various descriptive statistics, such as the mean, median, and mode. In skewed distributions, the median may be a better measure of central tendency than the mean, which can be heavily influenced by extreme values.

Skewness is an important consideration when selecting appropriate probability distributions for modeling data. Many common distributions, such as the normal, lognormal, or exponential distributions, have specific skewness properties that need to be accounted for.

**What is an asymmetric tail?**

A symmetric distribution like the normal distribution has no skewness. An asymmetric tail is when one tail of the distribution extends further than the other.

**What is the mean value?**

The arithmetic mean is calculated by dividing the sum of all values by the number of values.

For example, an array contains these values: 3,2,1

The sum is 3 + 2 + 1 equals 6

The number of values is 3.

6/3 equals 2. The average of 3, 2, 1 is 2

**What is a probability distribution?**

A probability distribution lets you analyze how likely different random values occurs, in other words, it shows how often we would expect to see different potential values.

**How to interpret the output from the SKEW function?**

Positive skewness implies a group of values with an asymmetric tail moving toward more positive values. Negative skewness implies a group of values with an asymmetric tail moving toward more negative values.

**Quartiles can also tell us about skewness**

Symmetrical distribution: Q_{2}- Q_{1} =Q_{3}- Q_{2}

Positively skewed: Q_{2}- Q_{1} < Q_{3}- Q_{2}

Negatively skewed: Q_{2}- Q_{1} > Q_{3}- Q_{2}

Learn more about quartiles: QUARTILE.INC | QUARTILE.EXC

**Box plots can also tell us about skewness**

Q1 (first quartile) - 25th percentile

Q2 (second quartile) - 50th percentile (median)

Q3 (third quartile) - 75th percentile

Positively skewed if the median is closer to Q1 than Q3. Negatively skewed if the median is closer to Q3 than Q1.

## 2. SKEW Function Syntax

SKEW(*number1, [number2], ...*)

## 3. SKEW Function Arguments

number1 |
Required. This argument can be an array of constants or a cell reference to a group of values. |

[number2] |
Optional. Up to 254 additional arguments separated by commas. |

## 4. SKEW Function Example 1

**A meteorologist is studying the distribution of daily rainfall measurements in a tropical region expecting lots of precipitation. They want to determine if the rainfall data is skewed, which could indicate the presence of extreme weather events or seasonal patterns. How can the SKEW function be used to assess the skewness of the rainfall data and potentially identify any deviations from a symmetric distribution?**

The data points are:

Data points | |||||

102 | 91 | 115 | 83 | 76 | 112 |

105 | 102 | 72 | 82 | 112 | 92 |

104 | 115 | 92 | 111 | 118 | 73 |

106 | 82 | 106 | 101 | 109 | 104 |

119 | 100 | 84 | 74 | 52 | 94 |

107 | 107 | 115 | 114 | 128 | 120 |

107 | 118 | 101 | 78 | 64 | 83 |

93 | 108 | 86 | 85 | 91 | 99 |

112 | 99 | 99 | 71 | 100 | 92 |

70 | 90 | 79 | 104 | 86 | 83 |

79 | 53 | 111 | 142 | 115 | 102 |

108 | 91 | 121 | 99 | 100 | 114 |

107 | 125 | 105 | 115 | 94 | 122 |

The argument is

*number1 = B25:G37*

The SKEW function allows cell references to cell ranges, in this particular example on cell range containing 78 observations.

Formula in cell B20:

The formula returns approx. -0.42. A negative skew value means that the distribution has a longer tail on the left side.

## 5. SKEW Function Example 2

**In an educational setting, teachers want to analyze the distribution of test scores for a particular subject. They suspect that the scores may be skewed due to the presence of a few exceptional students or potential grading issues. How can the SKEW function be applied to the test score data to identify and measure any skewness?**

The scores are:

Data points | |||||

35 | 60 | 47 | 36 | 71 | 28 |

22 | 74 | 56 | 58 | 83 | 49 |

94 | 76 | 80 | 89 | 46 | 82 |

67 | 7 | 44 | 62 | 92 | 37 |

55 | 54 | 15 | 66 | 30 | 17 |

31 | 36 | 61 | 45 | 56 | 16 |

46 | 43 | 32 | 83 | 12 | 50 |

44 | 78 | 70 | 29 | 25 | 52 |

55 | 81 | 63 | 46 | 30 | 21 |

47 | 31 | 39 | 40 | 46 | 22 |

66 | 34 | 42 | 65 | 59 | 60 |

74 | 30 | 30 | 78 | 35 | 34 |

65 | 9 | 32 | 37 | 89 | 21 |

The argument is

*number1 = B25:G37*

The SKEW function allows cell references to cell ranges, in this particular example a cell range containing 78 observations.

Formula in cell B20:

The SKEW function returns a value of approximately 0.19. A positive skewness value indicates that the distribution has an elongated tail extending towards the right or higher values.

In other words, when the SKEW function returns a positive value, it suggests that the distribution of the data is skewed to the right or positively skewed. This means that the tail on the right side of the distribution is longer or more stretched out compared to the tail on the left side.

A positive skewness value indicates that there are more extreme values or outliers on the higher end of the distribution causing the tail to extend further to the right. The majority of the data points are concentrated towards the lower values with a smaller proportion of data points trailing off towards the higher values.

## 6. SKEW Function Example 3

*In a study on household incomes, researchers want to analyze the distribution of incomes in a particular region. However, they suspect that the income data may be skewed due to the presence of a few extremely high or low incomes. How can the SKEW function be used to assess the degree of skewness in the income data?*

The income data is located in cell range B25:G37

The argument is

*number1 = B25:G37*

The SKEW function allows cell references to cell ranges, in this particular example a cell range containing 78 observations.

Formula in cell B20:

The SKEW function returns a value of approximately 0.003. A skewness value close to zero indicates that the distribution has symmetrical tails meaning the lengths of the tails on both sides of the distribution are roughly equal.

In other words, when the SKEW function returns a value very close to zero it suggests that the distribution of the data is approximately symmetric or bell-shaped. The tails on the positive and negative sides of the distribution extend to similar lengths without one tail being significantly longer or shorter than the other.

The closer the skewness value is to zero the more balanced and symmetrical the distribution is considered to be with the tails on both sides having roughly equal lengths or extensions. A skewness value of exactly zero would represent a perfectly symmetric distribution where the tails on both sides are identical in length and shape.

## 7. SKEW Function not working

The SKEW function returns #DIV/0! error value if there are less than three numbers to be calculated.

## 8. How is the SKEW Function calculated?

SKEW function = n/(n-1)(n-2)Σ(x_{i }- x̄/s)^{3}

n - number of data points

x_{i} - data point

x̄ - arithmetic mean

s - standard deviation

**What is the standard deviation?**

Standard deviation tells you how far from the average values are spread out. Both charts above have numbers and an average plotted, they share the same average however, the numbers are not the same.

Chart A above shows that the values are more spread out than the values in chart B. Chart A has a standard deviation of 23.45256334, standard deviation for chart B is 5.207075606. Standard deviation is used in statistics.

√(Σ(x - x̄)^{2}/n)

x̄̅ is the average.

n is how many values.

### Functions in 'Statistical' category

The SKEW function function is one of 73 functions in the 'Statistical' category.

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