Slope of a Line
In the graph on the next page, notice that the line has pitch, or slope, to it; this is also called “rise over run.” If you’ve ever done roofing, built a staircase, graded landscaping, or installed outflow piping, you’ve probably encountered the “rise over run” concept. If this graph were the roofline of a house we could talk about its pitch or rise over run—the ratio of the vertical distance (rise) divided by the horizontal distance (run) as we walk up the roof.
Using the following formula, we can calculate the slope (rise over run) of a straight line that goes through two points on a graph. The coordinates of the two points are (x1, y1) and (x2, y2).
The subscripts refer to the two points.
Example:This line has a slope of positive one.
This example shows a “perfect” relationship between y and x.Using an ozone monitor as an example, every time the true ozone level (x) goes up by 1 ppb, your instrument response (y) also goes up exactly the right amount…1 ppb.If your instrument went up 2 ppb instead of 1 ppb for every true rise in ozone level, the slope would be 2, meaning your instrument is giving a reading twice as high as it should be.
A line with a positive slope slants up and to the right.
Graph these coordinates (x, y). Using the equation above, calculate the slope of the line.
(1, 2)
(2, 4)
(3, 6)
Another way to calculate slope is to write the equation in terms of y giving us the standard equation for a straight line:
y = mx + b
The m is the slope(rise over run) and b is the y-intercept (the point where the line crosses the y axis on the graph). For example, if your ozone analyzer reads 2 ppb when the true ozone level is zero, then the y-intercept will be 2 ppb.
If we know the slope and the y-intercept, then for every value of x we can find the value of y. Likewise, for every value of y we can find the value of x. This is similar to using a calibration curve.
Practice:
In the following linear equations, identify the slope and the y-intercept.Make a small table. Choose some values for x(such as 1, 2, 3, 4, 5). Thenenter the values for y.
1)y = 2x + 3
2)y = x + 4
3)y = 3x + 1
y = 2x + 3Slope =
y-intercept =
x / y1
2
3
4
5
y = x + 4Slope =
y-intercept =
x / y1
2
3
4
5
y = 3x + 1Slope =
y-intercept =
x / y1
2
3
4
5
Now, using your x and y coordinates, draw the lines on graph paper.
Calibration
Calibration is the process of determining the relationship between a measurement standard and the output value on a measuring instrument. Calibration often involves adjusting the measurement instrument to agree with the value of the standard, within a specified accuracy. For example, a pH meter can be calibrated using standard pH buffers. The meter is adjusted so the digital read-out matches the value of the pH buffer.
Basically, a calibration curve IS A LINE and can be described by the equation for a line:
y = mx + b
In general, when you do a calibration you want your instrument readings to be as close as possible to the standard. Ideally, there will be zero difference between your instrument reading and the calibrationvalue.
The r-squared value is how close your line is to a straight line. An r-squared of 1 means that it is a perfectly straight line which of course never happens in real life. If your points are all over the place you might have an r-squared of 0.7 or even lower. (Excel can easily calculate your slope and intercept and r-squared values using the Excel functions of =SLOPE, =INTERCEPT, and =RSQ as shown in the screenshot and embedded Excel file below.)
Tip: if you want to plot a line in Excel always use a SCATTER CHART, not a line chart.
For many instruments, the calibration curve is linear through much of the instrument’s range. However, at the high and low ends of the range, the curve may cease to be linear.
Revised 5/20/14pe