Final answer:
The type of distribution can be symmetrical, skewed, or bimodal, and influences the decision between using the mean or median as the appropriate measure of center. The mean is calculated using midpoints of intervals and frequencies, and the median is based on the middle value or interval. A histogram helps identify the distribution's shape and thus the most suitable measure of center.
Step-by-step explanation:
To determine the type of distribution in a histogram, we examine the shape of the data. For a symmetrical distribution, the mean and median are usually close together, while for a skewed distribution, the mean and median will differ significantly. The mean is the arithmetic average of the data and is influenced by outliers or extreme values. The median is the middle value when the data are sorted in ascending order and is less affected by outliers. The mode is the most frequently occurring value in the data set.
To calculate the mean for grouped data, we find the midpoint of each interval and multiply it by the frequency of the interval. We then add all these products and divide by the total number of data values to approximate the mean. For the median, if the distribution is symmetrical, it will be the value in the middle interval. However, if the distribution is skewed, the median will be in an interval where the cumulative frequency reaches 50% of the total.
The most appropriate measure of center depends on the shape of the distribution. For symmetrical distributions, both the mean and median are suitable. For skewed distributions, the median is often most appropriate, as it is resistant to outliers.
A bimodal distribution will have two modes. The preferred measure for analyzing the center of data with outliers or extreme values is typically the median. The histogram helps identify the shape of the distribution, allowing for the selection of the appropriate measure of center and providing insight into the distribution's skewness.