STATISTICAL TREATMENT 233 



the median, and the mode. The arithmetic mean is the sum of all the 

 measurements divided by the number of measurements, and next to a 

 simple percentage is probably the most commonly used statistical con- 

 stant. The median is the point on the scale such that half the measure- 

 ments are above this point and half the measurements are below. It is 

 obvious that this definition is determinate if the number of observations 

 is odd and not too great, but if the number of observations is even or if 

 the number of measurements is great, the definition becomes indeter- 

 minate and some method of interpolation must be used. Such methods 

 will be illustrated later in connection with the use of probability paper for 

 representing a frequency distribution. The mode is defined as the point 

 on the scale where the maximum number of individuals occur, and 

 although this point may be determined directly from the observations, 

 it is subject to such a degree of sampling variation that it is probably 

 useless to treat the mode without some form of frequency curve discus- 

 sion. The difficulty of determination of the mode makes it the least 

 used of these three centering constants. 



As with the centering constant, the statistical indices of scatter are 

 numerous, the extent of the variation of the individuals from a centering 

 point being measured in terms of average deviation, standard deviation, 

 quartile limits, etc. Average deviation is just what its name indicates, 

 the average amount that the individuals deviate from the center, and 

 may be stated separately for positive deviations (above the center) 

 and negative deviations (below the center) or may be stated without 

 regard for the sign of the deviation. Standard deviation is the most 

 commonly used measure of scatter when the arithmetic mean is used 

 as the centering point. It is defined as the square root of the average of 

 the squares of the deviations of the individuals from the mean. Although 

 not so simple an index to grasp as the average deviation, it is used because 

 of its connection with the theory of errors and the normal probability 

 curve. Standard deviation is measured in the same unit as the mean, 

 that is, the yardstick of measurement, and it becomes a unit on this scale 

 in terms of which the variability of the indi^'iduals is measured. In 

 Table 3 has been presented the probability that an individual would 

 deviate from the mean by an amount equal to or greater than any 

 multiple of this unit of variation. These probabilities have been deter- 

 mined on the assumption that the material is distribvited according to 

 the normal probability curve, and Table 3 should be used only when this 

 assumption is reasonably well justified. The quartile limits are two points 

 on the scale of measurement of the same character as the median, and are 

 appropriately used in connection with the median as a measurement of 

 variation. The lower quartile is the point below which one-quarter of 

 the observations occur, and the upper quartile is the point below which 



