Final answer:
In a right-skewed distribution, the median is typically smaller than the mean because the higher values in the tail pull the mean upwards more than the median.
Step-by-step explanation:
When examining a skewed distribution, particularly one skewed to the right, understanding the relationship between the mean, median, and mode is crucial. In a right-skewed, or positively skewed distribution, the data points stretch out more towards the higher values on the right-hand side of the distribution graph, showing that there is a tail in the positive direction. This skewness indicates that there are several outliers or a long tail of values that are higher than the rest of the data.
In such cases, the mean is pulled towards these higher values more than the median or mode, resulting in the mean being greater than the median. Thus, if the graph is skewed to the right, the median is generally smaller than the mean. This can be seen in a right-skewed histogram where the right tail is longer, signifying that the mean is to the right of the median.
For example, consider the data set where the mean is 7.7, the median is 7.5, and the mode is seven. Here, the mean is the largest of the three measures of central tendency, indicating that the distribution is indeed skewed to the right. While the median is more resistant to outliers and represents the middle of the dataset, it doesn't capture the effect of these extreme values as much as the mean does.
To further illustrate, the mode is often the lowest value, the median falls in the middle, and the mean, due to its sensitivity to extreme values, tends to be the highest in a positively skewed distribution. This concept is valuable when analyzing data for understanding underlying trends and for making predictions based on those trends. An understanding of skewness is also essential when considering probability distributions and their applications.