Learn how to Read Exam charts
This article explores a range of statistical charts and their applications in quality control and process improvement. From Pareto charts highlighting key areas for error reduction to R&R studies assessing measurement accuracy, each chart provides unique insights for data analysis. It covers the use of Gage Linearity reports, Interaction Diagrams, P-Charts, and more to pinpoint process variations, control quality, and guide effective decision-making across industries.
1) Pareto
This chart is a Pareto chart, a type of graph that combines a bar chart and a line graph. It's typically used to prioritize potential causes of a problem because it shows both the individual and cumulative impact of different factors.
In this case, the chart is about Freight Billing Errors. The bar graph portion displays the number of errors for different types of defaults (e.g., Secondary Carrier, Missing Code, Incorrect Destination Postal Code, etc.). The heights of the bars indicate the number of errors for each type, with the leftmost bar (Secondary Carrier) having the most errors at 18, and thus being the tallest.
The line graph, which climbs from left to right, shows the cumulative percentage of the errors when the types are ranked in descending order. The first point on the line graph above "Secondary Carrier" shows that this type alone accounts for 41.9% of the errors. As you move right, you add the next category of errors, "Missing Code," to get a cumulative percentage. By the end of the chart, all categories combined reach 100%.
The purpose of the Pareto chart is to focus on the most significant factors, which in this context are the types of billing errors that occur most frequently. It suggests that efforts to reduce errors should first address "Secondary Carrier" and "Missing Code," as improving these areas will have the largest impact on reducing total errors.
2) R&R study
This chart is comparing the components of variation in a process or measurement system. It's a Gauge R&R study (Repeatability and Reproducibility), which is used to assess a measurement system's accuracy by determining the amount of variability introduced by the measurement system itself, versus the actual process variation.
Gauge R&R (Repeatability and Reproducibility): This bar represents the proportion of variation due to the measurement system (the gauge) itself. Repeatability refers to the gauge's consistency in measuring the same item multiple times, and reproducibility refers to the gauge's ability to remain consistent across different operators.
Repetitive: This likely refers to the variation observed when the same operator measures the same item multiple times.
Reproducibility: This is the variation when different operators use the same gauge to measure the same item.
Piece by Piece: This measures the variation from the actual process, when measuring different items using the same gauge and operator.
The chart is divided into three different types of variation, shown in different colors:
% Contribution: The percentage of total variation each component contributes to the overall process.
% Study Variation: The percentage of total study variation (which includes both the process and measurement system variations) that each component contributes.
% Process: The percentage of the process variation for each component when the measurement system is assumed to be perfect.
The purpose of this chart is to help understand which part of the measurement process is contributing the most to overall variability and to identify opportunities for improvement. For example, if Gauge R&R is a large proportion of the total variation, the measurement system may need to be improved. If the process variation is large, then improvements in the process might be necessary.
3) Gage Linearity and Bias Report
This image presents a Gage Linearity and Bias Report, which is a detailed analysis used to assess a measurement system's performance. Here's a breakdown of its components:
Gage Linearity Plot: The scatter plot on the left shows bias, which is the difference between the measured value and a known reference value, across the range of reference values. The dashed lines represent the confidence interval around the regression line. Ideally, you want the bias to be as close to zero as possible across all reference values, indicating that the measurement system is consistent and unbiased.
Gage Linearity Table: The table in the top right provides statistical details about the linearity of the gauge. A measurement system is considered linear if the bias does not change across the range of measurements. The "Constant" is the y-intercept of the linearity regression line, and the "Slope" should be close to zero for good linearity. The smaller the "S" (Standard Error of the estimate), the more precise the linearity estimate is. "R-Sq" represents the percentage of variation in bias that is explained by the linearity model; higher is better.
Gage Bias Table: This table gives the average measured bias at different reference values and the statistical significance (P-value) of that bias. A low P-value (typically < 0.05) indicates that the observed bias is statistically significant and not due to random chance.
Percent of Process Variation: The bar chart at the bottom right shows how much of the total process variation is due to linearity and bias. In a well-functioning measurement system, you'd want these to be as low as possible, indicating that the variability in your measurements is due to the process itself rather than problems with the gauge.
4) I-chart
This chart is an I-chart, a type of control chart used for monitoring the variation in individual values of a process over time. The chart plots individual measurements as a time series:
UCL (Upper Control Limit), the red line at 10.432, is the threshold that indicates the upper limit of expected variation for the process.
LCL (Lower Control Limit), the red line at 1.748, is the threshold that indicates the lower limit of expected variation.
X̄ (Process Mean), the green line at 6.045, represents the average of the individual measurements.
If the individual values remain within the control limits (between UCL and LCL), the process is considered to be in control. Points outside these limits would suggest that the process may be affected by special causes of variation and may need investigation. The process here appears to be in control since all points lie within the UCL and LCL.
5) Multi-Vari Chart
This chart is a Multi-Vari Chart, specifically comparing the hardness of butter as influenced by two factors: the type of agriculture (Industrial or Organic) and the temperature at which the butter is kept (Cold or Hot). The Y-axis represents the butter hardness in grams, and the chart plots individual data points for butter hardness at the two different temperatures, represented by different symbols, for each type of agriculture.
The chart shows that for both types of agriculture, butter is harder (has higher gram value) when cold and softer when hot, which is expected. However, the butter from industrial agriculture starts with a higher hardness when cold compared to organic, but decreases more significantly when hot. This suggests that the type of agriculture might influence how temperature affects butter hardness.
The dotted lines connecting the data points help to visualize the pattern or trend of the changes in butter hardness due to temperature changes within each agriculture type.
6) Normal Probability-Probability (P-P) Plot
This chart is a Normal Probability-Probability (P-P) Plot, used to assess if a dataset follows a normal distribution. The x-axis shows the observed weights of candies, and the y-axis shows the corresponding percentiles under the normal distribution.
In a P-P Plot, if the data are normally distributed, the points should fall approximately along the diagonal line which represents the expected normal distribution. In this plot, the points closely follow the line, suggesting the weight of the candies does closely follow a normal distribution.
The sidebar provides additional statistics:
Mean: The average weight of the candies is 0.9996 units.
Standard Deviation: The variability in the weight of the candies is 0.009957 units, which is quite low, indicating the weights are very consistent.
N: The sample size is 1,500 candies.
AD (Anderson-Darling): A test statistic of 0.503, which is used to measure how well the data follow a particular distribution. Smaller values generally indicate a better fit.
P-value: At 0.204, it is above the common significance level of 0.05, suggesting that there is not enough evidence to reject the null hypothesis that the data are normally distributed. This means the weight distribution of the candies can be considered normal.
7) Wilcoxon Rank test
The image shows results from three Wilcoxon Rank Tests, which are non-parametric tests used to compare medians from a single sample to a hypothesized median or to compare the medians of two matched samples.
Here's a breakdown of each test:
I. Wilcoxon Rank Test:
Hypothesis: The median of the data is different from 70.00 (two-tailed test).
N (sample size): 45
Test statistic: 320.0
P-value: 0.042
Estimated Median: 61.50
Since the P-value is less than 0.05, the result is statistically significant, and we would reject the null hypothesis that the median is 70. The estimated median of the data is 61.50.
II. Wilcoxon Rank Test:
Hypothesis: The median of the data is greater than 70.00 (right-tailed test).
N (sample size): 45
Test statistic: 320.0
P-value: 0.980
Estimated Median: 61.50
With a P-value of 0.980, there is no significant evidence to suggest that the median is greater than 70. The estimated median remains 61.50.
III. Wilcoxon Rank Test:
Hypothesis: The median of the data is less than 70.00 (left-tailed test).
N (sample size): 45
Test statistic: 320.0
P-value: 0.021
Estimated Median: 61.50
The P-value of 0.021 is less than the typical alpha level of 0.05, which indicates that there is significant evidence to suggest that the median is less than 70.00.
In summary, based on these tests, there is significant evidence to suggest that the median is not equal to and is in fact less than 70. There is no evidence to suggest that the median is greater than 70. The estimated median from the sample is 61.50.
8) Interaction Diagram
The image depicts an Interaction Diagram, which is a type of graph used in statistics to show how the level of interaction between two variables affects a third variable.
In this chart, the x-axis likely represents different levels or types of a categorical variable, although it's not explicitly labeled. The y-axis shows the average density of the pieces, which is the variable being influenced. The two lines likely represent two different temperatures of the oven as they interact with the categorical variable on the x-axis.
The points are labeled with numbers (515.0, 235.0, 223,000, and 267,000), which might represent the levels of the x-axis variable or another variable not labeled in the image. However, without additional context or labels on the x-axis, it's unclear what these numbers represent exactly.
The key takeaway from the interaction plot is that the relationship between the oven temperature and the average density of the pieces depends on the levels of the other variable. For one level (possibly at 515.0), the temperature does not have much effect on the average density, while for another level (at 235.0), the higher temperature is associated with a much higher average density. This suggests a significant interaction effect between the two variables on the density of the pieces.
9) Interaction Diagram
This image shows a Fitted Line Diagram, also known as a regression analysis chart. The chart includes:
Data Points: These are the blue dots representing observed values of the response variable for given doses.
Regression Line (in red): This line represents the best fit through the data points, based on the linear regression equation provided at the top (Response = Dose * 1.320 + 0.5105). This equation predicts the response for a given dose.
Confidence Intervals (95% CI, dashed purple lines): These lines provide a range for the estimated regression line, indicating where the true regression line is likely to exist with 95% confidence.
Prediction Intervals (95% PI, dashed green lines): These show where future observations are expected to fall with 95% confidence.
The values on the right indicate:
S (Standard Error of Estimate): 2.68365, which measures the spread of the data points around the fitted regression line. Lower values indicate a better fit.
R-Sq (Coefficient of Determination): 92.5%, suggesting that 92.5% of the variability in the response variable is explained by the dose.
R-Sq (adj): Adjusted R-Squared value of 91.4%, slightly adjusted for the number of predictors in the model to account for the diminishing returns of adding more predictors.
Overall, the diagram is used to assess the relationship between dose and response, showing a strong positive linear relationship with a high degree of explanation by the dose.
10) P-chart
The chart you provided appears to be a P-chart, which is a type of control chart used in statistical process control to monitor the proportion of defectives in a batch of items. Here's a brief explanation:
Y-Axis (Proportion): It shows the proportion of defective items in each subgroup.
X-Axis (Sub-Group): Each point represents a subgroup of observations.
Control Limits: The upper and lower "steps" represent the control limits, which are set based on statistical criteria. The top step is the Upper Control Limit (UCL), and the bottom step is the Lower Control Limit (LCL).
Central Line: The green horizontal line represents the overall average proportion of defects for all subgroups.
Data Points: The blue line connecting dots represents the proportion of defects in each subgroup.
Signal: The red marker indicates a potential signal or out-of-control point that may require investigation, as it lies beyond the Upper Control Limit (UCL).
In a P-chart, if data points are within the control limits, the process is considered in control. Points outside of the limits indicate that the process may be out of control, and special cause variation is present. The red point suggests that in that particular subgroup, the proportion of defectives is unusually high and may warrant further investigation.
11) X-bar & S chart:
The image shows two different types of statistical process control (SPC) charts:
The top chart is an X-bar chart, which is used to monitor the process mean over time. Each point represents the average of a set of measurements within a subgroup.
UCL (Upper Control Limit): 208.71, which is the calculated upper boundary of the process mean.
LCL (Lower Control Limit): 190.16, the calculated lower boundary.
Central Line (X̄): 199.44, which represents the overall process mean.
The bottom chart is an S-chart, which is used to monitor the variability of the process, specifically the sample standard deviation.
UCL (Upper Control Limit): 12.91, the upper boundary for variability.
LCL (Lower Control Limit): 0, the lower boundary. In some cases, LCL can be set to zero, particularly if the control limits are based on a distribution that does not include negative values.
Central Line (S): 5.70, which represents the average standard deviation across all subgroups.
Both charts are used to determine if a process is stable (in control) or if there are special causes of variation that need to be investigated. For the most part, if the points are within the upper and lower control limits, the process is considered to be in control. In these charts, all points are within the control limits, suggesting that the process is stable and in control.
12) I-MR:
The image shows an I-MR Chart, which consists of two related control charts used for process monitoring - specifically for when data is collected one at a time and the sample size is one.
Individuals Chart (I-Chart): The top chart monitors the process level over time.
UCL (Upper Control Limit): 113.98
Center Line (X-bar): 99.5, which represents the process average.
LCL (Lower Control Limit): 85.02
The red point indicates a single measurement that is an outlier, which might signify a special cause that should be investigated.
Moving Range Chart (MR-Chart): The bottom chart monitors process variability by plotting the difference between sequential measurements.
UCL (Upper Control Limit): 17.79, which represents the upper limit for the moving range values.
Center Line (MR): 5.44, which is the average moving range.
LCL (Lower Control Limit): 0. This is typical for an MR-chart since the range cannot be negative.
The red point on the MR-chart indicates a large shift in the value from the previous measurement, which could be a signal of a process change or an anomaly.
Both charts together help to determine if a process is stable (in statistical control) and to identify any out-of-control conditions that may occur. The Individuals chart assesses shifts in the process level, while the MR chart helps to identify changes in process dispersion. In this case, the process appears to be stable except for the one point marked in red in both charts, which may indicate a non-random change in the process at that time.
13) I-chart
The image shows an Individuals Control Chart, also known as an I-chart. This type of chart is used to monitor the variation in individual data points of a process over time. Here's the breakdown:
Y-Axis (Individual Value): Represents the measurement of the process at each time point or observation.
X-Axis (Observation): Represents the sequence of individual observations or time order.
Control Limits: The horizontal lines labeled 1 through 7 represent different statistical control limits, with Line 4 likely being the process mean. Lines 1 and 7 would be the Upper and Lower Control Limits (UCL and LCL, respectively).
Data Points: The connected dots are the individual measurements observed over time.
If all points fall between the control limits and there is no non-random pattern, the process is considered to be in control. The control chart helps to identify any signals of special cause variation, such as points outside the control limits or non-random patterns within the limits. In this chart, most of the data points fall within the control limits, but there is one point significantly higher than the rest, which could be an indication of a special cause that should be investigated.
14) Multi-Vari Chart
The chart you've provided is a Multi-Vari Chart, which is typically used to observe variations within a process and is often employed in quality control settings.
In this particular Multi-Vari Chart, we're looking at the variation in diameters of objects produced by two different machines under three different settings. Here's how to interpret the chart:
The x-axis (horizontal) represents "Settings," which likely correspond to different operational settings or conditions under which the machines are running. There are three settings labeled 1, 2, and 3.
The y-axis (vertical) shows the "Diameter" of the objects produced, which is the measurement of interest in this case.
There are two lines, each representing one of the two machines used in the process (Machine 1 and Machine 2). Machine 1 is denoted by circles (o), and Machine 2 by triangles (Δ).
Each point on the lines corresponds to the average diameter size produced by the machine under each setting. The line connects the average measurements to show the trend as the settings change.
By examining the chart:
For Machine 1, as the setting increases from 1 to 3, there is a trend of increasing diameter size. This could indicate that higher settings on Machine 1 lead to a larger diameter of the product.
For Machine 2, the diameter increases from setting 1 to 2, but then there's a sharp decrease at setting 3.
The difference in patterns between the two machines could suggest that they respond differently to the settings changes, or that there might be some inconsistency or issue with Machine 2 at setting 3.
The dashed red line might be indicating the overall trend when considering the data from both machines. However, given that the two machines have different patterns, this overall trend might not be very meaningful and could potentially mask the individual behavior of each machine.
The purpose of analyzing a Multi-Vari Chart is to determine the source of variation in the process. By comparing the outputs from the two machines across different settings, one can investigate whether the variation is consistent (common cause variation) or if there are specific changes that cause a shift in the diameter size (special cause variation).
15) Two sample T-test
The image is a summary output of a Two-Sample T-Test along with Confidence Interval (CI) analysis. This statistical test compares the means of two independent groups to see if there is a statistically significant difference between them. Here's a breakdown of the information presented:
Descriptive Statistics:
Group 1: Consists of 125 observations with a mean score of 1.800 and a standard deviation (StDev) of 0.700. The standard error of the mean (SE Mean) is 0.063.
Group 2: Has 150 observations with a mean score of 1.600 and a standard deviation of 0.900. The standard error of the mean is 0.073.
Estimation for Difference:
The difference in the mean scores between the two groups is 0.2000.
The 95% confidence interval for the difference in means is from 0.0999 to 0.3901. This interval suggests that we can be 95% confident that the true difference in means falls within this range.
T-Test:
The Null Hypothesis (H0): There is no difference in population means (μ1−μ2=0).
The Alternative Hypothesis (Ha): There is a difference (μ1−μ2=0).
T-Value: The calculated t-statistic is 2.07.
Degrees of Freedom (DF): The test has 271 degrees of freedom, which reflects the sample sizes minus the number of estimated parameters.
P-Value: The p-value is 0.039, which is the probability of observing such a difference (or more extreme) if the null hypothesis is true.
Interpretation:
The p-value is less than the typical alpha level of 0.05, indicating that there is statistically significant evidence to reject the null hypothesis. This means that there is a significant difference between the mean scores of Group 1 and Group 2.
The confidence interval does not contain zero, which also suggests that the difference in means is statistically significant.
The analysis assumes unequal variances, as stated at the beginning of the output.
In summary, this T-Test suggests that there is a significant difference between the scores of the two groups, with Group 1 having a higher mean score than Group 2.
16) comparing variances
The image you've uploaded appears to summarize the results of a statistical analysis comparing variances between two different production processes, "Sugar Apple Stinger Production" and "Tomato Spark". The summary includes confidence intervals for variance ratios, confidence intervals for standard deviations, and boxplots for each production process. Here's a breakdown:
95% CI for (σ of Sugar Apple Stinger Production) / (σ of Tomato Spark):
This part of the image provides the 95% Confidence Interval (CI) for the ratio of the standard deviations (σ) of the two production processes. A ratio of 1 would indicate that the variances are equal.
The Bonett's test and Levene's test are statistical tests for equality of variances. Both have P-values of 0.000, indicating that the null hypothesis of equal variances between the two groups is rejected at typical significance levels.
95% CI for StDevs:
This provides the 95% Confidence Intervals for the standard deviations of each process separately. Each blue line represents the range within which we can be 95% confident that the true standard deviation lies for each production process.
The confidence interval for "Sugar Apple Stinger Production" is much narrower than that for "Tomato Spark", suggesting more consistency in the production process of Sugar Apple Stinger.
Boxplot of Sugar Apple Stinger Production, Tomato Spark:
The boxplots give a graphical representation of the distribution of the production outputs.
Each boxplot shows the median (the line within the box), the interquartile range (IQR - the box), and the range (the "whiskers" extending from the box). Outliers can also be represented as individual points beyond the whiskers, though none are shown here.
The boxplot for "Sugar Apple Stinger Production" appears to show less variability and is more symmetric around the median compared to "Tomato Spark", which has a wider range indicating more variability.
Interpretation:
There is statistically significant evidence to suggest that the variances of the two production processes are different, as shown by the confidence interval for the variance ratio not including 1 and the P-values from both tests being 0.000.
The "Sugar Apple Stinger Production" process seems to have a smaller and more precise variability compared to "Tomato Spark".
Given the significant results from Bonett's and Levene's tests, it's clear that there are differences in the variability of the production processes that could be important for quality control and process improvement.
In summary, the tests suggest a significant difference in variability between the two production processes, and the boxplots visually support this finding by showing different distributions for each process.