Measuring Data Spread

Measures of data center, including the mean, median, and mode, provide useful information but do not tell the complete story of a dataset. Imagine two cities: City A has daily temperatures over a week of {18, 19, 20, 20, 21, 22, 20} degrees Celsius, and City B has {10, 30, 5, 15, 35, 25, 0} degrees Celsius. If you calculate the mean temperature for both, you'll find it's 20°C for both cities. However, the weather experience in these two cities is very different! City A is very consistent, while City B has wild temperature swings.

This is where measures of spread, also known as measures of dispersion or variability, become essential. They quantify how much the data points in your dataset tend to deviate from the average value. Let's look at the most common ways to measure this spread.

Range

The simplest measure of spread is the range. It's calculated by subtracting the minimum value in the dataset from the maximum value.

$$\text{Range} = \text{Maximum Value} - \text{Minimum Value}$$

For City A: Range = 22°C - 18°C = 4°C For City B: Range = 35°C - 0°C = 35°C

The range gives you a quick sense of the total span covered by your data. As you can see, City B has a much larger range than City A, reflecting its greater temperature variability.

While easy to calculate, the range has a significant drawback: it only considers the two most extreme values. A single very high or very low value (an outlier) can drastically affect the range, potentially giving a misleading picture of the overall data spread.

Variance

To get a more precise measure of spread that considers all data points, we use variance. Variance measures the average of the squared differences between each data point and the mean of the dataset. Squaring the differences serves two purposes:

1. It ensures all differences are positive (so negative and positive deviations don't cancel each other out).
2. It gives more weight to larger differences.

Speaking, for a dataset $x_1, x_2, ..., x_n$ with a mean $\bar{x}$, the variance involves these steps:

1. Calculate the mean ($\bar{x}$).
2. Find the difference between each data point and the mean: $(x_i - \bar{x})$.
3. Square each of these differences: $(x_i - \bar{x})^2$.
4. Calculate the average of these squared differences.

There are slightly different formulas for population variance (denoted $\sigma^2$, pronounced "sigma squared") and sample variance (denoted $s^2$). For introductory purposes, the concept is the main focus. A common formula for sample variance is:

$$s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$$

The division by $n-1$ instead of $n$ is a technical adjustment used when estimating the population variance from a sample, providing a better estimate.

Let's calculate the variance for City A (Mean $\bar{x} = 20$): Differences: (18-20), (19-20), (20-20), (20-20), (21-20), (22-20), (20-20) = {-2, -1, 0, 0, 1, 2, 0} Squared Differences: {4, 1, 0, 0, 1, 4, 0} Sum of Squared Differences: 4 + 1 + 0 + 0 + 1 + 4 + 0 = 10 Sample Variance $s^2 = 10 / (7-1) = 10 / 6 \approx 1.67$

Now for City B (Mean $\bar{x} = 20$): Differences: (10-20), (30-20), (5-20), (15-20), (35-20), (25-20), (0-20) = {-10, 10, -15, -5, 15, 5, -20} Squared Differences: {100, 100, 225, 25, 225, 25, 400} Sum of Squared Differences: 100 + 100 + 225 + 25 + 225 + 25 + 400 = 1100 Sample Variance $s^2 = 1100 / (7-1) = 1100 / 6 \approx 183.33$

As expected, the variance for City B (183.33) is much higher than for City A (1.67), indicating greater spread.

A limitation of variance is its units. If our original data was in degrees Celsius (°C), the variance is in degrees Celsius squared (°C²). This isn't very intuitive to interpret directly in relation to the original data.

Standard Deviation

This leads us to the most commonly used measure of spread: the standard deviation. It's simply the square root of the variance.

$$\text{Standard Deviation} = \sqrt{\text{Variance}}$$

For population standard deviation, it's denoted by $\sigma$ (sigma), and for sample standard deviation, it's $s$.

$$s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}$$

The primary advantage of the standard deviation is that it brings the measure of spread back into the original units of the data.

For City A: Standard Deviation $s = \sqrt{1.67} \approx 1.29$ °C For City B: Standard Deviation $s = \sqrt{183.33} \approx 13.54$ °C

Now we can say something more intuitive:

• In City A, the temperatures typically deviate from the mean (20°C) by about 1.29°C.
• In City B, the temperatures typically deviate from the mean (20°C) by about 13.54°C.

This clearly shows the much larger variability in City B's temperatures, expressed in understandable units.

A smaller standard deviation indicates that data points tend to be close to the mean (low variability, high consistency). A larger standard deviation indicates that data points are spread out over a wider range of values (high variability, low consistency).

This histogram compares the temperature distributions for City A and City B. Both have a mean of 20°C, but the spread of temperatures (indicated by the width of the distribution and the standard deviation, SD) is much larger for City B (orange) than for City A (blue).

Understanding these measures of spread - Range, Variance, and especially Standard Deviation - is fundamental for describing datasets. They provide insights into the consistency and variability within your data, which is often just as important as the average value.