For any process, it is important to know the variation pattern of the process in order to control and minimize the variations in the process. One of the ways to describe the process variation is to use tally chart and then configure a histogram.
Tally chart is a chart or a table that counts all the occurrences at each data value or each cell. The following table presents 125 observations on the inside diameter of forged piston rings used in an automobile engine. The data were collected in 25 samples of five observations each. Notice that there is some variability in piston ring diameter. However, it is very difficult to see any pattern in the variability or structure in the data in the table.
Table 1 Initial observations on the process
Sample Number Observations 1 74.030 74.002 74.019 73.992 74.008 2 73.995 73.992 74.001 74.011 74.004 3 73.998 74.024 74.021 74.005 74.002 4 74.002 73.996 73.993 74.015 74.009 5 73.992 74.007 74.015 73.989 74.014 6 74.009 73.994 73.997 73.985 73.993 7 73.995 74.006 73.994 74.000 74.005 8 73.985 74.003 73.993 74.015 73.988 9 74.008 73.995 74.009 74.005 74.004 10 73.998 74.000 73.990 74.007 73.995 11 73.994 73.998 73.994 73.995 73.990 12 74.004 74.000 74.007 74.000 73.996 13 73.983 74.002 73.998 73.997 74.012 14 74.006 73.967 73.994 74.000 73.984 15 74.012 74.014 73.998 73.999 74.007 16 74.000 73.984 74.005 73.998 73.996 17 73.994 74.012 73.986 74.005 74.007 18 74.006 74.010 74.018 74.003 74.000 19 73.984 74.002 74.003 74.005 73.997 20 74.000 74.010 74.013 74.020 74.003 21 73.988 74.001 74.009 74.005 73.996 22 74.004 73.999 73.990 74.006 74.009 23 74.010 73.989 73.990 74.009 74.014 24 74.015 74.008 73.993 74.000 74.010 25 73.982 73.984 73.995 74.017 74.013
A tally chart is configured in this page in terms of the frequency of the data, i.e., the number of occurrences at each dimension cell, as in Table 2.
Table 2. Tally chart for the observations in Table 1
Ring Diameter, x (mm) Tally Frequency Cumulative Frequency Relative Frequency Cumulative Relative Frequency73.965 <= x <= 73.970 1 1 1 0.008 0.00873.970 <= x <= 73.975 0 1 0.000 0.00873.975 <= x <= 73.980 0 1 0.000 0.00873.980 <= x <= 73.985 1111 111 8 9 0.064 0.07273.985 <= x <= 73.990 1111 1111 10 19 0.080 0.15273.990 <= x <= 73.995 1111 1111 1111 1111 19 38 0.152 0.30473.995 <= x <= 74.000 1111 1111 1111 1111 111 23 61 0.184 0.48877.000 <= x <= 74.005 1111 1111 1111 1111 11 22 83 0.176 0.66474.005 <= x <= 74.010 1111 1111 1111 1111 11 22 105 0.176 0.84074.010 <= x <= 74.015 1111 1111 111 13 118 0.104 0.94474.015 <= x <= 74.020 1111 4 122 0.032 0.97674.020 <= x <= 74.025 11 2 124 0.016 0.99274.025 <= x <= 74.030 1 1 125 0.008 1.000Total 125 1.000
A graph of the observed frequencies versus the piston ring diameter is shown in this page. This graph is called histogram. The tally chart and histogram present a visual display of the data in which one may more easily see three properties:
1. Shape of the distribution
2. Central tendency: the most probable value for the process
3. Dispersion or spread of the distribution.
Figure 1. Histogram of piston ring diameter
Application Example -- A use of tally chart to accomplish a cost saving:
Facts of the case:
In 100% inspection of completed electronic devices, a certain critical electrical characteristic was measured on a meter. This meter had a dial gage on which the value of the electrical characteristic was indicated by a pointer. Two fixed red lines were set on the dial, one at the maximum specified value and another at the minimum. This permitted the inspector to tell at a glance whether or not each device met the specifications, so that the inspector could rapidly sort the nonconforming product from the acceptable product.
Because this characteristic was responsible for a number of rejections and it was desired to study its pattern of variation, the inspector was asked to read the value registered on the dial gage for every device and to make a line on a frequency distribution tally chart indicating this value. Based upon the tally, a distribution bar diagram was created for each shift and sent immediately to the quality control engineer of the plant. These bar diagrams generally looked like Figure 2.
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Figure 2. Typical bar diagram showing frequency distribution of electrical characteristic of electronic device
One morning the distribution bar diagram for the preceding night shift looked like Figure 3 in the following page. This not only indicated a great increase in the proportion of nonconforming units but also showed an unusual distribution pattern. According to the tow tally charts, we could conclude that there is something wrong with the process.
Figure 3. This bar diagram called for investigation of the reasons for the unusual distribution pattern.
Analysis and action:
The quality control engineer, suspecting that this unusual distribution pattern might indicate an inspector error, went immediately to the inspection station and rescued from the scrap bin the devices rejected on the night shift. They were all retested by the day shift inspector, and nearly all proved to be satisfactory. It developed that there had been a new inspector on the night shift who had not understood how to operate the test equipment. Several hundred dollar worth of acceptable products were thus saved from the scrap heap. The new night shift inspector was instructed in the correct method of testing so that the mistake would not be repeated.
The following steps will be used to construct a tally chart and a histogram:
1. Count the number of observations (n).
2. Find the largest and smallest observations.
3. Find the range (the largest - the smallest).
4. Determine the number and width of the class intervals by the following rules:
a. Generally use from 4 to 20 cells.
b. Use class intervals of equal width. When choosing class intervals, be sure to choose values that leave no question of the interval in which a value falls.
c. As a rule of thumb, use for the number of class intervals.
d. Choose the lower limit for the first cell by using a value that is slightly less than the smallest data value.
d. The class interval h can be determined by h = range/ number of cells.
5. Tally the data for each cell determined in step 4 for a tally chart. The a histogram can be configured by drawing rectangular boxes with heights equal to the frequencies of the number of observations in each cell determined in step 4.
WORKING EXAMPLE
In a manufacturing production line that produces magnetic recording heads, a critical process parameter that is monitored to determine how the line is performing is gap width. In this process, 100 magnetic recording heads were examined to determine the gap width. The units of measurement of gap width are mm. The following table gives the results of this study.
Gap Width Readings for Magnetic Recording Heads 1.39 1.40 1.60 1.41 1.43 1.46 1.3 1.50 1.34 1.47 1.56 1.35 1.52 1.51 1.25 1.39 1.55 1.59 1.50 1.66 1.61 1.32 1.46 1.30 1.51 1.52 1.48 1.38 1.40 1.55 1.39 1.33 1.46 1.43 1.35 1.57 1.50 1.195 1.48 1.41 1.65 1.51 1.42 1.60 1.29 1.38 1.46 1.39 1.42 1.46 1.70 1.55 1.46 1.52 1.33 1.52 1.25 1.48 1.60 1.43 1.51 1.35 1.40 1.46 1.57 1.62 1.46 1.51 1.24 1.50 1.56 1.30 1.40 1.55 1.50 1.52 1.43 1.39 1.41 1.38 1.40 1.35 1.48 1.42 1.30 1.38 1.55 1.46 1.58 1.34 1.41 1.29 1.41 1.42 1.43 1.38 1.48 1.42 1.60 1.35
Make a tally chart in the following format and then draw a histogram for the process distribution.
Gap Width Cell Boundaries Tally Frequency Cumulative Frequency Relative Frequency Cumulative Relative Frequency 1.190 <= x < 1.242 . . . . . 1.242 <= x < 1.294 . . . . . 1.294 <= x < 1.346 . . . . . 1.346 <= x < 1.398 . . . . . 1.398 <= x < 1.450 . . . . . 1.450 <= x < 1.502 . . . . . 1.502 <= x < 1.554 . . . . . 1.554 <= x < 1.606 . . . . . 1.606 <= x < 1.658 . . . . . 1.658 <= x < 1.710 . . . . .
School of Technology
College of Business & Applied Sciences
Eastern Illinois University