Question

Indicate whether the five number summary corresponds most likely to a distribution that is skewed to the left, skewed to the right, or symmetric. (100,110,115,160,220)

Short answer

The distribution of the five number summary (100,110,115,160,220) is skewed to the right.

Step-by-step solution

Calculate Mean

Firstly, calculate the mean which is the sum of the given numbers (100,110,115,160,220) divided by the number of elements, which in this case is 5. The mean is: \((100+110+115+160+220)/5 = 141\).

Find Median

The median is given in the five-number summary. It is the middle number, which in this case is 115.

Identify Skewness

Now, compare the calculated mean (141) to the given median (115). Since the mean is greater than the median, you can determine that the distribution is skewed to the right.

Key concepts

Skewed Distribution

A skewed distribution is one in which the values of a dataset are not symmetrically distributed around the mean. The direction of the skewness is determined by the side of the distribution that has the longer tail. If the tail stretches out to the right, more towards the higher values, the distribution is said to be right-skewed or positively skewed. Conversely, if the tail extends to the left, more towards the lower values, it's considered left-skewed or negatively skewed.

In the context of the exercise, the five number summary provided: (100,110,115,160,220), hints at the distribution's shape. When a distribution is skewed to the right, the mean will typically be higher than the median because the longer tail of high-value outliers pulls the mean upwards. This is precisely what we observe in the step-by-step solution, where the mean is calculated to be 141, while the median, being the middle value, is identified as 115.

Mean and Median Comparison

The mean and median are both measures of central tendency, but they behave differently in the presence of skewed data. The mean is the arithmetic average, sensitive to extreme values or outliers. It's calculated by adding up all the numbers in a dataset and dividing by the count of those numbers. In contrast, the median is the middle value when the numbers are arranged in order, less affected by outliers or extreme values since it purely depends on the order of data points.

Comparing the mean and median can provide insight into the distribution's skewness. If the mean is significantly higher than the median, as in the exercise solution where the mean is 141 and the median is 115, it suggests a right-skewed distribution. Similarly, if the mean is lower than the median, it usually indicates a left-skewed distribution. This comparison is a quick and reliable method to gauge the asymmetry of a dataset.

Descriptive Statistics

Descriptive statistics summarize and describe the main features of a dataset. They offer a way to present large amounts of data in a simple and understandable format. The five number summary is a form of descriptive statistics that includes five critical values: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. These values give a quick overview of the data structure, showing its spread and central tendency.

The five number summary is instrumental in identifying the distribution's shape by illuminating the data's center, spread, and skewness. For instance, large gaps between quartiles and extremes can signal the presence of skewness. In our exercise, the five number summary of (100,110,115,160,220) suggests right-skewness — the median resides closer to the lower end of the dataset, and there's a significant jump from the third quartile (160) to the maximum (220). Descriptive statistics, such as the five number summary, are foundational in understanding the underlying characteristics of data in various fields, from academia to industry applications.