Labortory Exercise 2: Descriptive Statistics

Density Curve, Normal Quantile Plot

An accurate measure of the reaction time is of great importance for
the interpretation of experiments involving reactions of
subjects. Below 64 reaction times.

    28  22  36  26  28  28  26  24  32  30  27
    24  33  21  36  32  31  25  24  25  28  36
    27  32  34  30  25  26  26  25  23  21  30
    33  29  27  29  28  22  26  27  16  31  29
    36  32  28  40  19  37  23  32  29  24  25
    27  24  16  29  20  28  27  39  23

a. A histogram can show the features of a distribution that are
   obviously not normal, like peaks, clear skewness or gaps and clusters.
   Draw a histogram for the given times and combine them with a normal
   curve. Do you think the distribution is normal? What is the meaning of
   the total area under the curve? What is this area equal to?

b. Calculate mean and standard deviation.

c. A standardized observation z shows the distance of an original
   observation x to the mean in numbers of standard deviations,
   including the direction (positive means above, negative below the
   mean).  Assuming that the distribution is perfectly normal,
   validates the following formula:

       X - muu
   z = -------

   Convert the x-scores to the standardized z-score. What is the
   z-score for x=37?

d. Create a scatterplot from the data. Put the x-scores on the x-axis
   and the z-scores on the y-axis.

e. Sort the tabel so that the x-scores are arranged in size, in
   ascending order.  Next, add a new column and put in position
   numbers from 1 to 64. Which position is assigned to x=37?

f. Calculate all quantiles by dividing all sortingnumbers by the total
   amount of observations (64). Which quantile is assigned to x=37?

g. A quantile corresponding with a z-score (or nonstandardized: an
   x-score) gives the relative frequency of all values smaller or
   equal to z (or: x). Or: the chance getting a value smaller or equal
   to z (or: x). Described by formulas: P(Z<=z) (or: P(X<=x)).
   Look for the z-score of the quantile (or the P(Z<=z)) of x=37 in
   table A, and give that score.

h. In a normal-quantile-plot, each x-score is set out against the
   z-score.  The z-score is derived from the quantile that corresponds
   with the x-value (using table A).  In the graph, the x-scores are
   put on the x-axis, the z-scores on the y-axis. Draw the
   normal-quantile-plot. Combine this with the line z=x (this is the
   line connecting the dots of the scatterplot in d.). Is it a normal