Table of contents 1 Why is particle size important?
CENTRAL VALUES: MEAN, MEDIAN, MODE
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- SYMMETRIC DISTRIBUTION WHERE MEAN=MEDIAN=MODE
| CENTRAL VALUES: MEAN, MEDIAN, MODE
For symmetric distributions such as the one shown in Figure 2 all central values are equivalent: mean = median = mode. But what do these values represent? MEAN Mean is a calculated value similar to the concept of average. The various mean calculations are defined in several standard documents (ref.1,2). There are multiple definitions for mean because the mean value is associated with the basis of the distribution calculation (number, surface, volume). See (ref. 3) for an explanation of number, surface, and volume distributions. Laser diffraction results are reported on a volume basis, so the volume mean can be used to define the central point although the median is more frequently used than the mean when using this technique. The equation for defining the volume mean is shown below. The best way to think about this calculation is to think of a histogram table showing the upper and lower limits of n size channels along with the percent within this channel. The Di value for each channel is the geometric mean, the square root of upper x lower diameters. For the numerator take the geometric Di to the fourth power x the percent in that channel, summed over all channels. For the denominator take the geometric Di to the third power x the percent in that channel, summed over all channels. Understanding and interpreting particle size distribution calculations figure 2 | SYMMETRIC DISTRIBUTION WHERE MEAN=MEDIAN=MODE 3 The volume mean diameter has several names including D4,3. In all HORIBA diffraction software this is simply called the “mean” whenever the result is displayed as a volume distribution. Conversely, when the result in HORIBA software is converted to a surface area distribution the mean value displayed is the surface mean, or D 3,2. The equation for the surface mean is shown below . The description for this calculation is the same as the D4,3 calculation, except that Di values are raised to the exponent values of 3 and 2 instead of 4 and 3. The generalized form of the equations seen above for D4,3 and D3,2 is shown below (following the conventions from ref. 2, ASTM E 799, ) . Where: D = the overbar in D designates an averaging process (p-q)p>q = the algebraic power of Dpq Di = the diameter of the ith particle Σ = the summation of Dip or Diq, representing all particles in the sample Some of the more common representative diameters are: D10 = arithmetic or number mean D32 = volume/surface mean (also called the Sauter mean) D43 = the mean diameter over volume (also called the DeBroukere mean) The example results shown in ASTM E 799 are based on a distribution of liquid droplets (particles) ranging from 240 – 6532 µm. For this distribution the following results were calculated: D10 = 1460 µm D32 = 2280 µm D50 = 2540 µm D43 = 2670 µm These results are fairly typical in that the D43 is larger than the D50— the volume-basis median value. MEDIAN Median values are defined as the value where half of the population resides above this point, and half resides below this point. For particle size distributions the median is called the D50 (or x50 when following certain ISO guidelines). The D50 is the size in microns that splits the distribution with half above and half below this diameter. The Dv50 (or Dv0.5) is the median for a volume distribution, Dn50 is used for number distributions, and Ds50 is used for surface distributions. Since the primary result from laser diffraction is a volume distribution, the default D50 cited is the volume median and D50 typically refers to the Dv50 without including the v. This value is one of the easier statistics to understand and also one of the most meaningful for particle size distributions. 4 MODE The mode is the peak of the frequency distribution, or it may be easier to visualize it as the highest peak seen in the distribution. The mode represents the particle size (or size range) most commonly found in the distribution. Less care is taken to denote whether the value is based on volume, surface or number, so either run the risk of assuming volume basis or check to assure the distribution basis. The mode is not as commonly used, but can be descriptive; in particular if there is more than one peak to the distribution, then the modes are helpful to describe the mid-point of the different peaks. For non-symmetric distributions the mean, median and mode will be three different values shown in Figure 3. tải về 7.54 Mb. Chia sẻ với bạn bè của bạn:
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