The relationship between mean, median and mode is shown with these 3 histograms.

## 4 important observations after carefully examining the relationship between mean, median, and mode.

1. The mode **always** occurs where the frequency is highest.

Consider the following classes along with their frequencies to see why this is the case.

1-3 2

4-7 3

8-10 8

11-13 5

14-16 1

Now try to imagine what the raw data might look like. There are many options, but one of those options could be the one you see below.

2 3 4 7 5

8th **9** 10 8th **9**

10 8 **9** 11 11

13 12 13 14 15

16

As you can see, 9 (shown in dark blue) is a mode of this dataset and occurs with a frequency of 8 for the class. The reason for this is simple.

Note that each class can only have 3 values. Since the frequency is 8, you will also need to write 8 numbers (shown in light blue and dark blue).

Since you have to write 8 numbers, some numbers tend to repeat themselves.

2. For a symmetrical histogram, mean = median = mode (at least if they are not the same, they have values that are very close together).

Consider the classes along with their frequencies. This situation gives a symmetrical histogram.

1-3 2

4-6 3

7-9 6

10-12 3

13-15 2

The raw data could be

1 2 4 5 6 7 8 9

7 8 9 10 11 12 13 14

The modes are 7, **8th** and 9.

The mean is 7.875 and 7.875 is closely the same **8th**

The median is **8th**

You might conclude that mean = median = mode = 8

A left-skewed histogram is also known as a negatively skewed histogram, and a right-skewed histogram is also known as a positively skewed histogram.

3. Although it is not always predictable, for a **Histogram tilted to the left**The mean is the smallest, the mode is the largest, and the median is between the mean and the mode.

The mean is the smallest because outliers in the left tail pull the mean to the left. Also notice that there are more values on the left end than on the right end.

These values, including possible outliers, tend to pull the mean to the left and make its value smaller than the other measures of the middle.

4. Although it is not always predictable, for a **Right-sloping histogram**The mean is the largest, the mode is the smallest, and the median is between the mean and the mode.

The mean is the largest because outliers in the right tail pull the mean to the right. Notice again that there are more values on the right end than there are on the left end.

These values, including possible outliers, tend to pull the mean to the right and make its value greater than the other measures of the center.