We know something important: scores that are close to the mean are average in two ways

  They are neither very high nor very low.

  They are very frequent.

Scores that are much higher or lower than the mean are rare.

We can transform a distribution of raw scores into a distribution of z scores.  If  the raw score distribution is bell-shaped and symmetrical, we use the properties of the normal distribution to estimate the probability of observing different ranges of raw scores.  Click here (Mark-there is an error in this link)to construct a frequency histogram of both the conscientiousness data and the conscientiousness data transformed into z-scores.

   Click on the statements below which describe the z distribution of transformed conscientiousness scores? 

Now let's calculate the probability for some raw scores on the conscientiousness scale.   Because the conscientiousness distribution is not quite symmetrical, the probabilities computed using the normal distribution will be slightly different from the percentile ranks computed from the raw score distribution itself.  We will now examine how closely the standard normal probabilities approximate the empirically derived percentile ranks in value.  In order to do this, you will need to open two windows:  one that shows the frequency histogram for the raw score distribution and one that shows the standard normal distribution.

Click here (Mark-there is an error in this link)to create a frequency histogram, with an interval width of 1, for the raw score conscientiousness  distribution.  You can find the percentile rank of each raw score in the distribution by clicking on the bar in the histogram directly to the right of the bar over the targeted raw score or by clicking on the row in the table directly below the row containing the targeted raw score.

Click here to create a standard normal distribution.  Clicking on 'sampling distribution of the mean' under the 'sampling distributions' menu will open a window containing a standard normal distribution.  There are two arrows under the x-axis of the distribution.  Put the arrow at the left-hand end of the distribution at -5.0 (or minus infinity), and move the arrow at the right-hand end of the distribution until it points to the value of z you are interested in (values of z appear in yellow highlighting.  The area under the curve between minus infinity and  z  appears in blue and the probability of observing values of z between minus infinity and z is the numerical value in red at the top of the window.

The conscientiousness distribution has a mean of 24.10 and a standard deviation of 3.17.  To answer the questions below, use pencil and paper to compute the value of z for the raw score queried.  Find the probability in the standard normal distribution and multiply by 100.  Find the percentile rank in the frequency histogram.