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Data Intervals Help

By — McGraw-Hill Professional
Updated on Sep 13, 2011

Data Intervals

The following problems involve data intervals including quartiles, deciles, percentiles, and straight fractional portions. Let's consider an example involving climate change. (The following scenario is fictitious, and is for illustrative purposes only. It shouldn't be taken as actual history.)

The Distribution

Suppose you want to know if the average temperature in the world has increased over the last 100 years. You obtain climate data for many cities and towns scattered throughout the world. You are interested in one figure for each location: the average temperature over the course of last year, versus the average temperature over the course of the year one century ago. The term "a century ago" or "100 years earlier" means "100 years before this year (99 years before last year)."

In order to calculate a meaningful figure for any given locale, you compare last year's average temperature t, expressed in degrees Celsius (°C), with the average annual temperature s during the course of the year a century ago. You figure the temperature change, T, in degrees Celsius simply as follows:

T = ts

If t < s, then T is negative, indicating that the temperature last year was lower than the temperature a century ago. If t = s, then T = 0, meaning that the temperature last year was the same as the temperature a century ago. If t > s, then T is positive, meaning that the temperature last year was higher than the temperature a century ago.

Now imagine that you have obtained data for so many different places that generating a table is impractical. Instead, you plot a graph of the number of locales that have experienced various average temperature changes between last year and a century ago, rounded off to the nearest tenth of a degree Celsius. Suppose the resulting smoothed-out curve looks like the graph of Fig. 8-10. We could generate a point-by-point graph made up of many short, straight line segments connecting points separated by 0.18°C on the horizontal scale, but that's not what we've done here. Instead, Fig. 8-10 is a smooth, continuous graph obtained by curve fitting.

Data Intervals

Fig. 8-10. Illustration for Practice 1 through 7.

Practice 1

What do the points (–2,18) and (+2.8,7) on the graph represent?

Solution 1

These points tell us that there are 18 locales whose average annual temperatures were lower last year by 28°C as compared with a century ago, as shown by the point (–2,18), and that there are 7 locales whose average annual temperatures were higher last year by 2.88°C as compared with a century ago, as shown by the point (+2.8,7).

Practice 2

Suppose we are told that the temperature in the town of Thermington was higher last year by 1.38°C than it was a century ago. This fact is indicated by the vertical, dashed line in the graph of Fig. 8-10. Suppose L represents the proportion of the area under the curve (but above the horizontal axis showing temperatures) to the left of this dashed line, and R represents the proportion of the area under the curve to the right of the dashed line. What can be said about L and R?

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