Asymmetrical distribution, also known as skewed distribution, is a type of probability distribution where the values are not evenly distributed around the central value or mean. In other words, the distribution is not symmetrical, with the values being concentrated more on one side of the mean than the other. This can be represented visually in a histogram, where the distribution will be lopsided and will not have a symmetrical shape.

There are two types of asymmetrical distribution: positive skew and negative skew. In a positive skew, the values are concentrated more on the right side of the mean, with the tail of the distribution extending to the right. This means that there are a few values that are much higher than the rest, which pulls the mean to the right and makes the distribution skewed. An example of a positively skewed distribution is the income of a group of people, where a few people may have extremely high incomes, while the majority have lower incomes.

In a negative skew, the values are concentrated more on the left side of the mean, with the tail of the distribution extending to the left. This means that there are a few values that are much lower than the rest, which pulls the mean to the left and makes the distribution skewed. An example of a negatively skewed distribution is the height of a group of people, where a few people may be very short, while the majority are taller.

It is important to note that asymmetrical distribution does not mean that the data is not normal or that it is not following a certain probability distribution. In fact, many distributions can be skewed and still be considered normal. For example, the normal distribution can be skewed if the data has a few extreme values that are pulling the mean to one side.

Asymmetrical distribution can have significant implications for statistical analysis and decision making. For instance, the mean and the median may not be the same in a skewed distribution, and the choice of which measure of central tendency to use can depend on the skewness of the data. In a positively skewed distribution, the mean will be pulled to the right by the extreme values, while the median will be less affected and will be closer to the center of the distribution. In this case, the median may be a better measure of central tendency. On the other hand, in a negatively skewed distribution, the mean will be pulled to the left by the extreme values, and the median will again be less affected and closer to the center of the distribution. In this case, the mean may be a better measure of central tendency.

Another example of the implications of asymmetrical distribution is in the calculation of the standard deviation. The standard deviation is a measure of how spread out the values in a distribution are, and it is calculated by taking the square root of the variance. In a symmetrical distribution, the mean, the median, and the mode are all the same, and the standard deviation is a meaningful measure of how spread out the values are around the center of the distribution. However, in a skewed distribution, the mean, the median, and the mode may not be the same, and the standard deviation may not accurately reflect the dispersion of the values.

In conclusion, asymmetrical distribution is a type of probability distribution where the values are not evenly distributed around the central value or mean. It can be represented visually in a histogram, and it can have significant implications for statistical analysis and decision making. There are two types of asymmetrical distribution: positive skew and negative skew, and the choice of which measure of central tendency to use and how to calculate the standard deviation can depend on the skewness of the data.