The meaning of symmetry and kurtosis

Sample Solution

Symmetry

Symmetry in a distribution refers to the way the data is distributed around the mean. If a distribution is symmetrical, the mean, median, and mode will all be equal and the data will be evenly distributed on either side of the mean.

Asymmetry

Asymmetry in a distribution refers to the lack of symmetry in the distribution. If a distribution is asymmetrical, the mean, median, and mode will not be equal and the data will not be evenly distributed on either side of the mean.

Full Answer Section

Bias

Bias in a distribution refers to the tendency of the mean to be pulled towards one side of the distribution. There are two types of bias: positive bias and negative bias.

Positive bias occurs when the mean is pulled towards the right side of the distribution. This can happen if there are a few extreme values on the right side of the distribution.

Negative bias occurs when the mean is pulled towards the left side of the distribution. This can happen if there are a few extreme values on the left side of the distribution.

Kurtosis

Kurtosis in a distribution refers to the peakedness or flatness of the distribution. There are three types of kurtosis: mesokurtic, leptokurtic, and platykurtic.

Mesokurtic distributions are the most common type of distribution. They have a bell-shaped curve and the mean, median, and mode are all equal.

Leptokurtic distributions are more peaked than mesokurtic distributions. They have a higher peak and fatter tails.

Platykurtic distributions are flatter than mesokurtic distributions. They have a lower peak and skinnier tails.

The meaning of symmetry and kurtosis

Symmetry refers to the shape of the distribution, while kurtosis refers to the peakedness of the distribution. A symmetrical distribution is one in which the data is evenly distributed on either side of the mean. A positively skewed distribution is one in which the data is pulled towards the right side of the mean. A negatively skewed distribution is one in which the data is pulled towards the left side of the mean. A leptokurtic distribution is one in which the data is more peaked than a mesokurtic distribution. A platykurtic distribution is one in which the data is flatter than a mesokurtic distribution.

Examples of the types of these shape measures

Here are some examples of the types of symmetry and kurtosis:

• A symmetrical distribution with a mesokurtic shape would be a normal distribution.
• A positively skewed distribution with a leptokurtic shape would be a distribution of income, where there are a few very wealthy people and many people with lower incomes.
• A negatively skewed distribution with a platykurtic shape would be a distribution of test scores, where there are a few very low scores and many scores that are close to the mean.

Importance of recognizing the type of symmetry and kurtosis in analyzing results of descriptive measures in decision making

The type of symmetry and kurtosis can be important in analyzing the results of descriptive measures in decision making. For example, if a distribution is negatively skewed, the median may be a better measure of central tendency than the mean. If a distribution is leptokurtic, the standard deviation may not be a good measure of variability.

In general, it is important to be aware of the type of symmetry and kurtosis of a distribution when interpreting descriptive measures. This can help to ensure that the results are interpreted correctly and that the right decisions are made.