Measures of dispersion capture the variability of the data. They provide context for interpreting a measure of central tendency. While the central measure (i.e., mean, median, or mode) is the most representative value for the data, the measure of dispersion provides more information about how the values in the data set differ from that central point. We can only assess the dispersion for numerical data that is at least an ordinal level of measurement.

There are four main measures of dispersion:

- Range
- Interquartile Range
- Variance
- Standard Deviation

Use the tabs above to learn more about each of these measures.

The **range **is the difference between the highest value in the dataset and the lowest value in the dataset. To compute the range, simply subtract the smallest value from the largest value. Consider the following example:

12 17 21 13 11 15 17 18 20 17 11

**Largest value: **21

**Smallest value: **11

**Difference: **21 - 11 = 10

The range for this dataset is 10.

The range can be used for any numerical data, but provides more information for data that does not contain outliers. With outliers, the range will reflect the extreme values, but may not accurately reflect a majority of the data.

The **interquartile range** (IQR) is a special type of range that is computed using the quartiles instead of minimum and maximum values. To compute the IQR, determine the first quartile and the third quartile values. Then, subtract the first quartile value from the third quartile value. Consider the following example:

12 17 21 13 11 15 17 18 20 17 11

** Step 1**: Order the numbers from least to greatest.

11 11 12 13 15 17 17 17 18 20 21

** Step 2**: Identify the quartiles. (Hint: use the median approach to divide the list in half and then divide each half in half)

11 11 12 13 15 17 17 17 18 20 21

**First Quartile: **12

**Third Quartile: **18

**Difference: **18 - 12 = 6

The interquartile range for this dataset is 6.

The interquartile range is a good alternative to the range when there are outliers or if there is skewness in the distribution. Since the interquartile range reflects the range for the middle 50% of the data, it is less subject to influence by outliers and extreme values.

We are going to discuss variance and standard deviation together because of their direct connection. **Variance **is a measure of how much individual values vary about a mean. This value has little real-world interpretation. However, when we take the square root of that variance, we get the **standard deviation**. The standard deviation can be interpreted as the average amount of deviation of each point from the mean. Thinking about this in simple terms, it's like finding the difference between each point and the mean and then computing the average of those differences. And this is exactly what we see when computing the variance and standard deviation. Let's consider the process for computing standard deviation using the following dataset:

12 17 21 13 11 15 17 18 20 17 11

The table below depicts the steps for computing variance and standard deviation. Note that the table below shows values rounded to four decimal places, but actual calculations were done with unrounded values. If you are using rounded values in your calculations, you may get a slightly different answer. Check with your instructor to see what their rounding guidelines are for rounding values before the final answer.

** Step 1**: Place the list of values in a vertical column

*This is computing the

Step 1 | Step 3 | Step 4 | |
---|---|---|---|

12 | 12 - 15.6364 = -3.6364 | (-3.6364)^2 = 13.2231 | |

17 | 17 - 15.6364 = 1.3636 | (1.3636)^2 = 1.8595 | |

21 | 21 - 15.6364 = 5.3636 | (5.3636)^2 = 28.7686 | |

13 | -2.6364 | 6.9504 | |

11 | -4.6364 | 21.4959 | |

15 | -0.6364 | 0.4050 | |

17 | 1.3636 | 1.8595 | |

18 | 2.3636 | 5.5868 | |

20 | 4.3636 | 19.0413 | |

17 | 1.3636 | 1.8595 | |

11 | -4.6364 | 21.4959 | Step 6 = 122.5455/11= 11.1405 |

Step 2 = 15.6364 |
Step 5 = 122.5455 |
Step 7 = √11.1405= 3.3377 |

The variance/standard deviation is appropriate to report if you're reporting the mean as the measure of central tendency.

- Last Updated: Sep 29, 2024 3:52 PM
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