QUIZ 4 DESCRIPTIVE STATISTICS: SHAPE OF DATA-SKEWNESS & KURTOSIS

This histogram shows

• skewed to the left.

• the right-hand side seems “chopped off” compared to the left side.

• the skewness is negative.

• a positive measure of skewness.

Answer explanation

From the figure:

1. Data:

4; 5; 6; 6; 6; 7; 7; 7; 7; 8

2. Mean is 63/10=6.3

4. Mode is 7

Which statement is TRUE about this histogram?

• Skewed to the right.

• Symmetrical.

• Data is skewed to the left.

• The mode is often less than the median, which is less than the mean.

Answer explanation

Data: 6; 7; 7; 7; 7; 8; 8; 8; 9; 10

1. Mean is = 77/10=7.7

2. Median is between 5th and 6th data = (7+8) /2 =7.5

3. Mode is 7

= mean > median > mode

If the distribution of data is skewed to the left

• the mean is less than the median.

• the mean is often less than the mode.

• the mode is often less than the median.

• the mode is less than the mean.

Answer explanation

Skewed to the left:

mode >median>mean = mode the largest

If the distribution of data is skewed to the right

• the mode is less than the median.

• the mode is less than the mean.

• the mode is more than the median.

• the mode is more than the mean.

Answer explanation

Skewed to the right:

mean >median> mode= mean the largest

Skewness

• is the degree of distortion from the symmetrical bell curve or the normal distribution.

• has three types: positive, zero and negative.

• measures the lack of symmetry in data distribution.

• is all about the tails of the distribution.

Answer explanation

Skewness has two types: positive and negative.

Kurtosis is all about the tails of the distribution

Skewness

• is a measure of the symmetry in a distribution

• is equal to 0 for a symmetrical dataset.

• measures the relative size of the two tails.

• is zero for a normal distribution.

Kurtosis

• measures the degree to which a given distribution is more or less ‘peaked’ relative to the normal distribution.

• has 3 categories of distributions:

  Leptokurtic

  Mesokurtic

  Platykurtic

• all about the tails of the distribution

• measure of outliers present in the distribution.

If a distribution has negative kurtosis 

• indicates that the data exhibit less extreme outliers than a normal distribution.

• has a flatter peak compared to a normal distribution.

• it is said to be leptokurtic.

• it is said to be platykurtic.

Answer explanation

Negative kurtosis has a flatter peak and thinner tails compared to a normal distribution.

If a distribution has positive kurtosis

• is known to be leptokurtic.

• has a sharper peak compared to a normal distribution.

• has thinner tails compared to a normal distribution.

• mean more data values are located on the tails.

Answer explanation

Positive kurtosis has sharper peak and heavier tails compared to a normal distribution.

A leptokurtic distribution

• has less values in the distribution tails

• has heavier tails than a normal distribution.

• has a higher peak than a normal distribution.

• has negative kurtosis value.

Answer explanation

1. has a positive kurtosis value.

2. has more values in the distribution tails.