From the
Exploring Data website - http://curriculum.qed.qld.gov.au/kla/eda/
© Education Queensland, 1997
Using Statistics in
Human Movements
One measure of form for
a runner is stride rate, defined as the number of steps per
second. A runner is considered to be efficient if the stride rate
is close to optimum. The stride rate is related to speed; the
greater the speed, the greater the stride rate.
In a study of 21 top
female runners, researchers measured the stride rate for
different speeds. The following table gives the average stride
rate of these women versus the speed.
Source: R.C. Nelson, C.M. Brooks, and N.L.
Pike, Biomechanical comparison of male and female runners,
in P. Milvy (ed.), The Marathon: Physiological, Medical,
Epistemiological, and Psychological Studies, New York Academy of
Sciences, 1977, pp. 793-807.
| Speed |
15.86
|
16.88
|
17.50
|
18.62
|
19.97
|
21.06
|
22.11
|
| Stride
Rate |
3.05
|
3.12
|
3.17
|
3.25
|
3.36
|
3.46
|
3.55
|
1. Plot the data on a
scattergram. Decide if the data appears to be linear.
2. Find the equation of
the least squares regression line.
3. Interpret the
gradient of this function. In other word, what are the
‘units’ on the gradient?
4. Plot the least
squares regression line on the scattergram.
5. Make a prediction of
the stride rate if the speed is 19 feet per second.
6. What is the stride
rate if the speed is 0? Interpret your result. Do the results
make sense? What should be done about this?
Extension
Do a residual plot of
the data. Do you notice any patterns? Can you find a better
mathematical model for this data?
Using Statistics in Human
Movements - A Solution Using the TI-83 Graphical Calculator
1. Enter the following data into
a Statistics Lists, say L1 and L2.
| Speed |
15.86
|
16.88
|
17.50
|
18.62
|
19.97
|
21.06
|
22.11
|
| Stride Rate |
3.05
|
3.12
|
3.17
|
3.25
|
3.36
|
3.46
|
3.55
|
2.
Construct a scatterplot from this data.
|
|
| 3.
Use (Zoom, 9:ZoomStat) to resize the scatterplot window
automatically. The
data appears to follow a linear relation quite nicely.
|
|
| 4.
To get the regression equation, press:
(Stat,Calc,4:LinReg(ax+b)), then (L1, L2,
Vars,Y-Vars,1:Function,1:Y1, Enter). This gives the gradient and
y-intercept of the regression equation. In this example
we are finding the relationship between two rates so the
units on the gradient are a bit tricky. The units are
(steps per second) per (feet per second). Real data can
be messy!
5. Use Graph to see the
scatterplot and the graph of the regression line.
The regression line fits
the data extremely well.
|
|
| 6.
To use the fitted values to predict the stride rate for a
speed of 19 feet per second, press (Vars,Y-Vars,
Function,Y1,(19),Enter). The predicted stride rate for a
speed of 19 feet per second is 3.29.
The predicted stride rate
for a speed of 0 feet per second is 1.766, which is
silly! Obviously, the model doesn’t work very
effectively for all speeds. A decision needs to be made
about the domain for which the model is valid.
|
|
Extension
| 7.
Extension - Fitting residuals to the scatterplot. The residual for each data value
is found by the formula
Residual = Data Value -
Fitted Value
A scatterplot of
residuals helps to determine the validity of the linear
model. Ideally, the residuals should show no strong
patterns.
To get a list of the
residuals, put the cursor at the top of list L3, and
press (List,RESID,Enter). Now do a scatterplot of List L3
verus List L1.
The scatterplot shows the
data has a strongly curved pattern. So another model, say
a quadratic one, may give a slightly better fit.
Should a more complicated
model be used? In this case there is no single correct
answer to this question.. A quadratic function may be
slightly more accurate, but with a loss of simplicity.
In this case, the linear
model is certainly acceptable and would have a
correlation coefficient close to 1. My decision would be
to use the linear model, and note in a report that the
residual plot does have a strong underlying pattern, and
that this indicates that the data might better be
modelled by a non-linear function.
|
|