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When interpreting data, the range is one of the simplest measures we can use to understand the spread. The range is quite simply the difference between the largest and smallest values in a dataset. It helps us get a quick sense of the variability. For example, in our dataset gathered from Pomona, we identified that the highest air quality index is 60, and the lowest is 28. Therefore, the range is calculated by subtracting the smallest number 28 from the largest number 60, resulting in a range of 32. This indicates that the air quality indices spread is 32 points from lowest to highest. The simplicity of the range doesn't account for how the values are distributed in-between the smallest and largest values, so it's often paired with other statistics like interquartile range for a fuller picture.

The interquartile range (IQR), offers a more resistant measure of variability than the range. It considers the spread of the middle 50% of data, filtering out potential outliers. To compute the IQR, we first need to divide our dataset into quartiles. The first quartile (Q1) marks the 25th percentile, and the third quartile (Q3) the 75th percentile. In our sorted Pomona dataset, Q1 is 45 and Q3 is 55. Thus, the IQR is calculated as the difference between the third and first quartiles:

• Q3 - Q1 = 55 - 45, leading to an IQR of 10.

This measure is particularly useful because it shows that the middle 50% of air quality index values in Pomona are spread out over a range of just 10 points, highlighting a concentrated central dispersion.

Sample variance is a step further in understanding data variability. It gives us insights into how each data point differs from the mean value. It takes into account all the data points and measures the average squared deviations from the mean. In our case, after calculating the mean of the Pomona dataset as approximately 48.33, we calculate each data point's deviation from this mean, square it, and average those squared deviations. The formula for sample variance is:

• \[ s^2 = \frac{(28 - 48.33)^2 + (42 - 48.33)^2 + ... + (50 - 48.33)^2}{9 - 1} \]
• The sample variance is approximately 103.07, which indicates the extent to which the values deviate from the mean. A larger variance signifies that the numbers are more spread out.

The sample standard deviation is closely related to the variance and provides a more intuitive measure of spread by returning to the original unit of measurement. This value is simply the square root of the sample variance. For the Pomona dataset, we calculated the standard deviation to be roughly 10.15. Given that the variance was 103.07, taking the square root simplifies this back to the unit of the data values.

• \[ s \approx \sqrt{103.07} \approx 10.15 \]
• This measure indicates how much the actual values of the dataset deviate, on average, from the mean value of the data. A standard deviation of 10.15 suggests that on average, each air quality index value is 10.15 points away from the mean of 48.33, giving us a tangible sense of the variability.

Compared to Anaheim with a higher standard deviation of 11.66, Pomona's air quality readings are relatively less variable.