trend line

Examples of trend line in the following topics:

• Line of Best Fit

  • The trend line (line of best fit) is a line that can be drawn on a scatter diagram representing a trend in the data.
  • The trend line, or line of best fit, is a line that can be drawn on a scatter diagram representing a trend in the data.
  • Trend lines typically are straight lines, although some variations use higher degree polynomials depending on the degree of curvature desired in the line.
  • This graph will be used in our example for drawing a trend line.
  • Illustrate the method of drawing a trend line and what it represents.

• Testing the Significance of the Correlation Coefficient

  • If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
  • If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
  • If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.
  • Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.
  • Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

• Beginning with straight lines

  • While the relationship is not perfectly linear, it could be helpful to partially explain the connection between these variables with a straight line.
  • Straight lines should only be used when the data appear to have a linear relationship, such as the case shown in the left panel of Figure 7.6.
  • The right panel of Figure 7.6 shows a case where a curved line would be more useful in understanding the relationship between the two variables.
  • We only consider models based on straight lines in this chapter.
  • If data show a nonlinear trend, like that in the right panel of Figure 7.6, more advanced techniques should be used.

• Conditions for the least squares line

  • The data should show a linear trend.
  • If there is a nonlinear trend (e.g. left panel of Figure 7.13), an advanced regression method from another book or later course should be applied.
  • The variability of points around the least squares line remains roughly constant.
  • 7.11: The trend appears to be linear, the data fall around the line with no obvious outliers, the variance is roughly constant.

• Types of outliers in linear regression

  • The secondary cloud appears to be influencing the line somewhat strongly, making the least square line fit poorly almost everywhere.
  • (5) There is no obvious trend in the main cloud of points and the outlier on the right appears to largely control the slope of the least squares line. (6) There is one outlier far from the cloud, however, it falls quite close to the least squares line and does not appear to be very influential.
  • You will probably find that there is some trend in the main clouds of (3) and (4).
  • In (5), data with no clear trend were assigned a line with a large trend simply due to one outlier (!).
  • Points that fall horizontally far from the line are points of high leverage; these points can strongly influence the slope of the least squares line.

• Plotting Lines

  • A line chart is often used to visualize a trend in data over intervals of time – a time series – thus the line is often drawn chronologically.
  • A line chart is typically drawn bordered by two perpendicular lines, called axes.
  • In statistics, charts often include an overlaid mathematical function depicting the best-fit trend of the scattered data.
  • It is simple to construct a "best-fit" layer consisting of a set of line segments connecting adjacent data points; however, such a "best-fit" is usually not an ideal representation of the trend of the underlying scatter data for the following reasons:
  • In either case, the best-fit layer can reveal trends in the data.

• Introduction to fitting a line by least squares regression

  • The lines follow a negative trend in the data; students who have higher family incomes tended to have lower gift aid from the university.
  • This video covers important ideas and consideration pertaining to fitting a straight line to data.

• Describing linear relationships with correlation

  • The correlation is intended to quantify the strength of a linear trend.
  • It appears no straight line would fit any of the datasets represented in Figure 7.11.
  • We'll leave it to you to draw the lines.
  • In general, the lines you draw should be close to most points and reflect overall trends in the data.
  • The first row shows variables with a positive relationship, represented by the trend up and to the right.

• Introduction to line fitting, residuals, and correlation

  • The equation for the line is
  • In each case, the data fall around a straight line, even if none of the observations fall exactly on the line.
  • The first plot shows a relatively strong downward linear trend, where the remaining variability in the data around the line is minor relative to the strength of the relationship between x and y.
  • The second plot shows an upward trend that, while evident, is not as strong as the first.
  • The last plot shows a very weak downward trend in the data, so slight we can hardly notice it.

• Displaying Data

  • Newspapers and the Internet use graphs to show trends and to enable readers to compare facts and figures quickly.
  • Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar chart, the histogram, the stem-and-leaf plot, the frequency polygon (a type of broken line graph), pie charts, and the boxplot.
  • In this chapter, we will briefly look at stem-and-leaf plots, line graphs and bar graphs.