Question

If the graph is skewed to the left, is the median bigger or smaller than the mean?

Answer 1

In a left-skewed distribution, the median is typically larger than the mean due to the influence of lower values in the tail of the distribution.

Step-by-step explanation:

When analyzing the distribution of data on a graph and it is skewed to the left, the median is often larger than the mean. This pattern is due to the nature of how outliers and a non-symmetrical distribution can influence the average value (mean) of the dataset. In a left-skewed distribution, there is a longer tail on the left side of the distribution, which contains lower values that pull the mean towards it, making it smaller than the median.

In contrast, for symmetric distributions, like the normal distribution, the mean, median, and mode all coincide at the same central location on the graph. However, when dealing with skewed data, these measures of central tendency separate, with the mean moving towards the direction of the skew. It's important to note that the mean is more susceptible to extreme values compared to the median, hence why in a left-skewed graph, the mean is less than the median.

Examples in practical datasets demonstrate this concept, such as when examining certain economic data where a majority of values are on the higher end, but there are substantial low-end outliers, resulting in a left skew. In such cases, the mean will reflect the influence of these low-end values more than the median will. To get a more accurate representation of the central tendency in a skewed distribution, it is better to look at the median or graphically represent the data using a histogram or box plot.

Answer 2

Answer: bigger

Explanation

A bell curve is symmetric with both tails mirror copies of each other. The mean and median are equal for any symmetric bell curve.

Now imagine stretching out the left tail so it's much longer. This makes the graph skewed to the left (aka negatively skewed).

These smaller outliers will pull on the mean to make the mean smaller. The median is robust and won't move (very much). Even if it did move, it won't move very much compared to how the mean moves.

Therefore mean < median in distributions that are skewed to the left.

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Let's try out an example.

Consider the set {81,82,83,84,85}. The mean and median are 83. The two items are equal in this symmetric distribution.

Now let's say we replaced 81 with 7 and the set is now {7,82,83,84,85}

The mean and median are 68.2 and 83 respectively. The median has not changed. But the mean has decreased. The small outlier (7) has pulled on the mean to make it smaller. This has mean < median and the distribution is skewed to the left.