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Section 2.4 Representing Data

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Objectives Create graphs to reveal patterns in data. Interpret graphs.

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Graphs When data is analyzed, scientists are often looking for patterns. Displaying data in a TABLE (or chart) may or may not reveal a pattern. Creating a GRAPH, however, helps to reveal patterns that may exist.

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**Graphs A graph is a visual display of data.**

27.37% Na A graph is a visual display of data. There are different types of graphs. A circle graph or pie chart is useful to show parts of a whole. 57.14% O 14.30% C

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Graphs A bar graph is used to show how a quantity varies across categories. The quantity being measured (dependent variable) appears on the y-axis and the categories (independent variable) appears on the x-axis.

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Graphs A line graph represents the points of intersection of data for 2 variables. The independent variable is plotted on the x-axis and the dependent variable is on the y-axis. Line graphs are used to the exclusion of all others in chemistry.

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Line Graphs Because data points can be very scattered, a line connecting all of them cannot be drawn. A line must be drawn so that the same number of points fall above & below the line. This is called a BEST-FIT line.

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Creating a Line Graph Two (straight) axes are drawn on graph paper and labeled. The x-axis represents the variable that is manipulated (changed) while the y-axis represents the variable that responds to the changes in the manipulated variable. Each label must include the name of the variable AND the unit attached to that variable.

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Creating a Line Graph A scale for each axis must be selected and assigned. Determine the highest & lowest values of the data. Subtract the values to find the range of the data. (NOTE: If you want the axis to start at 0, you must use 0 as the lowest value!) Count the number of boxes along the axis. Divide the range value by the number of boxes to find the value that will be represented by each box. Assign the determined scale to the axis, labeling clearly with the appropriate values. It may not be necessary to label every box but label enough boxes so that the scale is evident.

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**Creating a Line Graph Important Points to Note:**

The scale for each axis must be determined separately. Each axis, therefore, will have its own scale and units. They can, but do not have to be, the same. A “break in the scale” symbol can NEVER be used on a line graph in Chemistry. Choose a scale that is easy to plot & easy to read. The scale and labels must be consistent along the axis – the value of each “box” must remain the same. The scale cannot be changed as the axis is labeled. Scales do not have to start at zero. It is very important, therefore, to assign a value for each axis at their start. A properly scaled axis will not have any data points that fall “off-scale”. If you find this happens, you must re-scale your axis.

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**Creating a Line Graph Data points are plotted on the graph.**

The values of the independent & dependent variables form ordered pairs (x,y). The ordered pairs can be plotted on the graph from their corresponding x-axis or y-axis. Dependent Variable Y-axis (x, y) y x Independent variable X-axis

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**Creating a Line Graph A “best-fit” line is drawn.**

A straight “best-fit” line is almost always drawn. Sometimes a “best-fit” curve is needed. Instructions must be read carefully to determine which to draw. “Dot-to-dot” lines are NEVER drawn. Remember, the data points do not need to be on the line or curve drawn. In fact, sometimes none of the data points will actually be on the line/curve drawn.

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**Interpreting a Line Graph**

Calculating Slope A straight line always has a constant slope. 2 points ON THE LINE (NOT data points) must be chosen: (x1, y1) & (x2, y2) These points can be used with this formula to calculate slope: y2- y1 or Δy x2-x Δx Note: when reporting the slope, include the units (y/x)!

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**Interpreting a Line Graph**

What does the slope tell you? If a slope can be determined (that is, the line is straight), there is a linear relationship between the variables. If the line slopes upward, the slope is positive. This means the dependent (y) variable increases as the independent (x) variable increases. This represents a DIRECT relationship. If the line slopes downward, the slope is negative. This means the dependent (y) variable decreases as the independent (x) variable increases. This represents an INVERSE relationship.

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**Interpreting a Line Graph**

If a slope cannot be determined (that is, the “line” is curved), there is a nonlinear relationship between the variables. As can be seen in this graph, as the independent variable (x) increases, the dependent variable (y) decreases. This curve, therefore, shows an inverse relationship.

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**Interpreting a Line Graph**

Extrapolation When a “best-fit” line is drawn, the points on the line are considered continuous data points. That means you can read data from any place along the line. If the line that was drawn is extended beyond the plotted points, it has been extrapolated. Estimate values for the variables can be read from an extrapolated line.

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Practice Problems Plot the data in each table. Explain whether the graphs represent direct or inverse relationships. Table Table 2 Effect of Pressure on Gas Effect of Pressure on Gas Pressure (mm Hg) Volume (mL) 3040 5.0 1520 10.0 1013 15.0 760 20.0 Pressure (mm Hg) Temperature (K) 3040 1092 1520 546 1013 410 760 273

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