-1 B ASIC M EASUREMENT C ONCEPTS D ATA I N T E R P R E TAT I O N

1 3 -1 B ASIC M EASUREMENT C ONCEPTS D ATA I N T E R P R E TAT I O N

Data interpretation and representation are very important when working with on-line process control operations. Measurement devices provide the control system with important information about the inner workings of the process. Therefore, every user must clearly understand what data is being collected by the measurement devices and how it should be interpreted. This will help the user to apply the control program correctly, so that the process will behave in a predictable manner.

To understand the data-gathering process, you must first understand how instrumentation and data-collecting devices interpret data readings. These devices can interpret data sampling readings four different ways, each with

a different meaning. These methods for interpreting information include:

• mean • median • mode • standard deviation

Mean. The mean is the average value of a set of readings. This value is useful in applications that require an estimation of future or expected readings. To illustrate the mean, let’s use an instrument that emits a signal at set time intervals (every 10 seconds). This signal ranges from 2 to 20 mV and represents the mean value of the measurements taken during the 10-second time interval. That is, each signal’s value is the average of the readings taken since the last signal. Let’s suppose that the instrument last emitted a 13 mV signal and that it will send another signal in 10 seconds. Meanwhile, the instrument collects data every two seconds, resulting in values of 14 mV, 14.5 mV, 15 mV, 14.7 mV, and 14.8 mV. The mean of these readings, expressed

as X , is defined as: Industrial Text & Video Company 1-800-752-8398

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+ X 2 + X 3 +… X n

n or

Therefore, at the new reporting time, the instrument will emit a 14.6 mV signal—the mean value of the five readings (i.e., n = 5):

14 mV + 14.5 mV + 15 mV + 14.7 mV + 14.8 mV

5 = 14.6 mV

Median. The median is the middle value of a set of readings that are organized in ascending order. The following equations define the median:

() 2 + 1 for an odd number of samples

X m + X m () 2 () 2 + 1

for an even number of samples

2 where: M = the median

m = the total number of readings

X = a reading value

The readings from the previous example placed in ascending order are 14 mV,

14.5 mV, 14.7 mV, 14.8 mV, and 15 mV. This is an odd number of values (i.e., m = 5). Therefore:

M = X 51

2 () + = X 3

The value X 3 is the third value in ascending order, so the median is 14.7 mV. Note that for an even number of samples the median is the mean of the two center values.

The median calculation provides statistical information about the data measurements taken and is more tolerant of errors than the mean calculations. Referencing the previous example, if the 14 mV reading had been erroneously

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read as 20 mV due to noise in the system, the mean would have increased to

15.8 mV, whereas the median would have remained 14.7 mV. Thus, the median value is not greatly affected by extreme deviations caused by measurement errors.

Mode. The mode is the most frequent value or values in a set of data. The mode value for the following set of instrumentation readings—14 mV, 14.5 mV, 14 mV, 14.5 mV and 14.5mV—is 14.5 mV, because it is the most frequent value. If six readings had been taken and the sixth one was 14 mV, two mode values would have existed, 14.5 mV and 14 mV (three occurrences of each).

Mode values occur mostly in discrete processes, where events are not broken down into infinitesimal readings, as in analog processes. PLC count readings from analog input modules rarely contain a significant mode value, since the continuous nature of the signal constantly introduces changes into the readings. Therefore, the mode is not as valuable as the mean and median in determining measurement errors.

Standard Deviation. Often, an application requires information not only about the mean value of a set of process readings, but also about how these readings are distributed in relation to the mean. The standard deviation provides valuable information about a group of data, thus aiding in the quantitative evaluation of the sample measurements.

To demonstrate standard deviation, let’s examine a set of five instrument readings: 9 mV, 9.5 mV, 15 mV, 19.7 mV, and 19.8 mV. The mean of this sample is 14.6 mV, yet the readings are very dispersed. Standard deviation measures the spread of these values in relation to the mean and is expressed as:

n = 1 to σ= i n − 1

where: σ = the standard deviation

X = the calculated mean n = the number corresponding to each reading,

starting at 1 and ending at the last reading, i This formula computes the deviation of each sample from the mean. The

larger the standard deviation value, the more spread out the values (samples) Industrial Text & Video Company 1-800-752-8398

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are from the mean. For our instrument readings, the standard deviation will have a value of σ = 5.26:

When the data values are evenly distributed around the mean in a bell form, they are said to have a normal distribution or Gaussian distribution (see Figure 13-1). The standard deviation in a Gaussian (normal) distribution measurement provides information that allows for a quantitative determina- tion about how the data is spread. In a normal distribution, several conclu- sions can be obtained:

• 68% of all readings lie within ±1σ (see Figure 13-2a) • 95% of all readings lie within ± 2 σ (see Figure 13-2b) • 99% of all readings lie within ± 3 σ (see Figure 13-2c)

Figure 13-1. Normal distribution curve.

For example, if a set of reactor vessels in a continuous process has two temperature control loops, one that maintains a 358 ° C temperature with a standard deviation of 40 ° C and another that maintains a 358 ° C temperature with a standard deviation of 20 °

C, we know that the latter provides us with

a more peaked graph about the mean. In fact, 68% of the temperature readings in the second loop lie between 338 ° C and 378 °

C, while in the first loop, 68%

of the readings lie between 318 ° C and 398 ° C.

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X X+3 σ Figure 13-2. Distribution of data values as a function of standard deviation.

X–3 σ