Welding professionals often collect and analyze data on weld quality, defect rates, and production parameters to improve their processes. Presenting this data accurately enables better decision-making and quality control in fabrication projects. This chapter focuses on methods to present welding-related data effectively by understanding different types of variables and the organization of data sets.

4.1 Data presentation

Raw measurements from a welding workshop, such as bead widths, hardness readings or counts of defects, are hard to read as a long list. Data presentation arranges them in tables and graphs so that patterns, typical values and unusual results show up at once. The choice of table or graph depends on the type of variable: a continuous variable can take any value in a range (bead width, current, temperature), while a discrete variable takes separate values, usually counts (number of pores, number of weld passes). It also depends on whether the data is left as individual values (ungrouped) or sorted into class intervals (grouped).

Example: a quality supervisor measures the width of 40 fillet weld beads from one shift. The list of 40 numbers tells little, but a grouped table and a histogram show the typical width, the spread and any bead that is far from the rest.

4.1.1 Continuous variables

A continuous variable is measured, not counted, and can take any value within a range, limited only by the accuracy of the instrument. Bead width in millimetres, tensile strength in newtons per square millimetre, welding current in amperes and the temperature of a plate are all continuous. Continuous data is usually grouped into class intervals and drawn as a histogram, or plotted against time as a line graph.

4.1.1.1 Histogram

A histogram shows the distribution of continuous data. The horizontal axis is the variable divided into class intervals and the bars touch each other, because there is no gap between one class and the next. For equal class widths, the height of each bar is the frequency.

Example: the widths in millimetres of 40 fillet weld beads were measured in the order below.

6.3  5.7  6.7  6.2  6.7  6.7  5.6  5.8  6.7  6.8
7.1  6.1  7.7  5.7  6.0  4.5  6.9  7.6  7.3  4.5
7.0  6.2  4.6  5.4  5.7  6.1  5.3  5.4  6.8  6.3
6.6  7.0  7.7  5.9  5.9  7.0  8.3  7.0  7.5  6.9

Steps to draw the histogram:

1. Find the smallest and largest values: 4.5 mm and 8.3 mm. The range is 8.3 − 4.5 = 3.8 mm.
2. Choose the number of classes, usually between 5 and 10. With 5 classes the width must be at least 3.8 ÷ 5 = 0.76 mm, so a convenient width of 1.0 mm is used.
3. Write the classes so that each value belongs to one class only: 4.0 to under 5.0, 5.0 to under 6.0, and so on.
4. Count how many values fall in each class by going through the list once and marking each value as it is counted. The frequencies must add up to 40.
5. Draw the axes: classes along the bottom, frequency up the side, with an even scale starting at zero. Draw a bar for each class with no gaps, and give the graph a title and labels with units.

Frequency
16 |           ████
15 |           ████
14 |           ████
13 |           ████
12 |           ████
11 |           ████
10 |      ████ ████ ████
 9 |      ████ ████ ████
 8 |      ████ ████ ████
 7 |      ████ ████ ████
 6 |      ████ ████ ████
 5 |      ████ ████ ████
 4 |      ████ ████ ████
 3 | ████ ████ ████ ████
 2 | ████ ████ ████ ████
 1 | ████ ████ ████ ████ ████
   +--------------------------
     4-5  5-6  6-7  7-8  8-9    Bead width (mm)

The tallest bar is the 6.0 to under 7.0 mm class, so the largest group of beads, 16 of the 40, is about 6.5 mm wide. The bars fall away on both sides, and one bead in the 8.0 to under 9.0 mm class stands apart from the rest and should be checked.

4.1.1.2 Line

A line graph shows how a continuous variable changes, usually with time. The points are plotted at their measured values and joined by straight lines, because values between two readings also exist. It is used for temperature, current or voltage readings recorded at regular intervals.

Example: the temperature of a plate was measured every 5 minutes after the last pass.

Steps:

1. Draw time on the horizontal axis and temperature on the vertical axis, each with an even scale and labels with units.
2. Plot each pair of values as a point.
3. Join the points with straight lines.
4. Give the graph a title.

Reading between points gives an estimate. At 12 minutes the temperature lies between 245 °C at 10 minutes and 210 °C at 15 minutes. The fall in 5 minutes is 35 °C, so in 2 minutes it is about 2 ÷ 5 × 35 = 14 °C, and the estimate is 245 − 14 = 231 °C. The line falls steeply at first and then flattens, which shows that the plate cools fastest when it is hottest.

4.1.2 Discrete variable

A discrete variable takes separate values, usually whole numbers from counting: the number of pores in a weld, the number of passes in a joint, the number of rejected joints in a day. There are no values between 2 pores and 3 pores. Discrete data is displayed in frequency tables, bar graphs and pie graphs, and the pore counts below are used for the frequency table and the bar graph, and a count of defect types for the pie graph.

Example: an inspector counted the pores visible on the radiograph of each of 30 welds.

0  4  1  0  1  1  5  3  1  2
4  1  2  2  2  0  0  0  1  1
0  2  2  4  3  3  1  3  1  2

4.1.2.1 Bar graph

A bar graph uses separate bars of equal width, with gaps between them to show that the values are separate. The height of each bar is the frequency. A bar graph is also used for categories, such as defect types, in which case the order of the bars does not matter.

Steps: label the horizontal axis with the values (0, 1, 2, ... pores), scale the vertical axis from zero in even steps, draw each bar with the same width and a gap, and add a title.

Number of welds
 9 |        ████
 8 |        ████
 7 |        ████   ████
 6 | ████   ████   ████
 5 | ████   ████   ████
 4 | ████   ████   ████   ████
 3 | ████   ████   ████   ████   ████
 2 | ████   ████   ████   ████   ████
 1 | ████   ████   ████   ████   ████   ████
   +-------------------------------------------
      0      1      2      3      4      5    Pores per weld

The graph shows that one pore per weld is the most frequent result, and that welds with four or five pores are few.

4.1.2.2 Pie graph

A pie graph shows how a total is divided between categories. The whole circle is 360°, and each category gets a sector whose angle is its share of the total:

Angle = (frequency ÷ total) × 360°

Example: the 50 defects found in one month were counted by type.

To draw it, draw a circle, mark a starting radius, measure each angle in turn with a protractor, label each sector with its name and share, and give the chart a title. Check that the angles add up to 360°. The largest sector is porosity, so the workshop should look first at shielding gas coverage and surface cleanliness.

4.1.3 Grouped data

When there are many values, they are sorted into class intervals and the number of values in each class, the frequency, is recorded. A grouped frequency table is shorter and easier to read than the raw list, but the individual values are no longer shown. These terms are used:

• Class interval: a range of values, such as 6.0 to under 7.0 mm.
• Class boundaries: the lower and upper limits of a class (6.0 and 7.0).
• Class width: upper boundary minus lower boundary (7.0 − 6.0 = 1.0 mm).
• Class mark (midpoint): the centre of the class, (6.0 + 7.0) ÷ 2 = 6.5 mm.
• Frequency: the number of values in the class.
• Relative frequency: the frequency divided by the total, often shown as a percentage.

Example: the bead widths of the earlier 40 welds, with class marks and relative frequencies.

4.1.3.1 Histogram

A histogram of a grouped table is drawn with the class boundaries on the horizontal axis and touching bars. When the classes have equal width, the height of the bar is the frequency. When the class widths are different, the area of the bar must be proportional to the frequency, so the height is the frequency density:

Frequency density = frequency ÷ class width

Example: the time taken to complete a root pass on 54 joints was grouped into classes of unequal width.

The bar heights are 1.2, 2.4, 2.0 and 0.8, and the widths are 5, 5, 10 and 20. The area of the third bar is 10 × 2.0 = 20, which equals its frequency. If frequency had been plotted directly, the 30 to under 50 minute class would have looked as tall as the others, although it spreads its 16 joints over 20 minutes.

4.1.3.2 Bar

A bar graph of grouped data is used when the data is discrete and has been grouped, for example the number of pores per weld grouped as 0 to 1, 2 to 3 and 4 to 5. The bars have gaps between them, because the groups contain whole numbers only, unlike the touching bars of a histogram.

Example: the 30 pore counts from the discrete example, grouped.

Draw three separate bars of heights 15, 11 and 4 over the group labels. The graph shows that half of the welds have at most one pore (15 of 30), while 4 of 30 welds, about 13%, have four or five pores.

4.1.3.3 Cumulative frequency

The cumulative frequency of a class is the number of values up to the upper boundary of that class. It is found by adding each frequency to the total of the frequencies before it. The last cumulative frequency equals the total number of values.

The table answers questions such as: how many beads are narrower than 7.0 mm? The answer is 29 of 40, which is 72.5%.

4.1.3.4 Ogive

An ogive is the graph of cumulative frequency. The points are plotted at the upper class boundaries (the cumulative frequency up to a value cannot be known until the whole class is counted) and joined by a smooth curve or by straight lines. The curve starts at zero at the lower boundary of the first class.

Points for the bead widths: (4, 0), (5, 3), (6, 13), (7, 29), (8, 39) and (9, 40).

Reading values from the ogive:

• Median: go across from half the total, 40 ÷ 2 = 20, to the curve, then down to the axis. The 20th value lies between (6, 13) and (7, 29). By proportion it is 6 + (20 − 13) ÷ (29 − 13) × 1 = 6 + 7 ÷ 16 = 6.44 mm.
• Lower quartile: the value at 40 ÷ 4 = 10. It lies between (5, 3) and (6, 13): 5 + (10 − 3) ÷ 10 = 5.7 mm.
• Upper quartile: the value at 3 × 40 ÷ 4 = 30. It lies between (7, 29) and (8, 39): 7 + (30 − 29) ÷ 10 = 7.1 mm.
• Interquartile range: 7.1 − 5.7 = 1.4 mm. Half of the beads have widths within a range of 1.4 mm.

4.1.4 Ungrouped data

Ungrouped data is kept as individual values. When the list is long and values repeat, a frequency table is used, with one row for each different value. Because no class intervals are used, no information about individual values is lost, which makes ungrouped data suitable for discrete variables with a small number of different values and for small data sets.

Example: eight hardness readings (HV) taken along a weld, in the order of testing: 152, 158, 149, 163, 155, 158, 151, 160.

4.1.4.1 Line

A line graph of ungrouped data plots each reading against the order in which it was taken, joining the points. It shows trends and sudden jumps in a series of readings, and a horizontal line at the mean shows how far each reading is from the average.

Plot the points (1, 152), (2, 158), (3, 149), (4, 163), (5, 155), (6, 158), (7, 151) and (8, 160), with the test number on the horizontal axis and the hardness on the vertical axis. The mean is (152 + 158 + 149 + 163 + 155 + 158 + 151 + 160) ÷ 8 = 1 246 ÷ 8 = 155.75 HV. Draw a dotted line at 155.75 across the graph: readings 2, 4, 6 and 8 lie above it and readings 1, 3, 5 and 7 below it. The readings alternate and show no steady rise or fall.

A second kind of line graph, the line graph of frequencies, draws a vertical line at each value with a height equal to its frequency. For the pore counts it has lines of heights 6, 9, 7, 4, 3 and 1 at 0, 1, 2, 3, 4 and 5 pores.

4.1.4.2 Cumulative frequency

For ungrouped data, the cumulative frequency of a value is the number of readings that are less than or equal to that value. It is found by adding the frequencies in order.

From the table, 22 of the 30 welds, 22 ÷ 30 = 73.3%, have at most two pores. Because the 15th reading is the last of the 1-pore welds and the 16th is the first of the 2-pore welds, the median of the 30 readings is (1 + 2) ÷ 2 = 1.5 pores. Plotting the cumulative frequency against the value as a step graph, with a horizontal step at each value, gives the ungrouped version of the ogive.

Chapter Summary

This chapter covered the presentation of data using different graphical methods suited for continuous and discrete variables, including histograms, line graphs, bar graphs, and pie charts, as well as techniques for grouped and ungrouped data such as cumulative frequency curves and ogives. It then explored measures of central tendency, detailing how to calculate the mean, mode, and median for both grouped and ungrouped data sets. The chapter proceeded to discuss measures of dispersion, focusing on the calculation of standard deviation and variance, again distinguishing between grouped and ungrouped data. In the probability section, methods for calculating probabilities with and without replacement were explained. The chapter also introduced probability distribution functions, specifically the binomial and Poisson distributions, highlighting their applications. Finally, the normal distribution was examined as a fundamental continuous probability distribution, emphasizing its properties and relevance in statistical analysis. This comprehensive coverage equips learners with essential skills for applying statistics and probability in practical contexts.