In a "perfect" bell curve,what is the relationship between the mean, median, and mode? What Z score would " go with" each one?

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ncchemist | eNotes Employee

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A bell curve is a Gaussian function that graphically depicts the distribution of a range of data.  It is called a bell curve due to the fact that it resembles the general shape of a bell with the apex in the center, curving downward rapidly, and then flaring out at the bottom.  The salient point of the bell curve is that the most common data points are concentrated near the center with only a very small number deviating greatly to either side (these points are called outliers).

Let's define the mean, median, and mode.  The mean is basically the numerical average (sum all of the numbers and divide by the number of data points).  The median is the central number in a group of numbers dividing the upper half from the lower half.  Finally, the mode is the most commonly found number in a data set.  In a perfectly shaped bell curve, it is quite simple to locate these values.  They are all found at the same point on the bell curve: the highest point in the center of the curve.  This peak value is the average (mean), the center of the curve (median), and the most frequently occurring number (mode).

The z score is another term for the standard score, which is the number of standard deviations a particular value on the bell curve is located away from the mean value (or center of the curve).  The standard deviations are calculated by a particular formula, but suffice it to say that approximately 68% of the values will have a z score of 1 or less, 95% will have a z score or 2 or less, and 99.5% will have a z score of 3 or less.  Since the mean, median, and mode all lie at the center of the curve with no standard deviation, they all three have a z score of 0.

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