Histogram

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Raw Data


Purpose

The purpose of a Histogram is to depict the frequencies of numerical or measurement data in a bar chart.

The tool represents one of the means of visualizing the shape, arithmetic mean and dispersion of a distribution along the scale of measurement.

Histograms, or frequency plots, show the distribution of the data by displaying how often different values occur. They are used to summarize data from a process in graphical form. The graph helps answer the question of whether the process is capable of meeting customer requirements.


Anatomy

how to construct a histogram

Reference: Juran's Quality Control Handbook

Terminology

A. Vertical axis - Scale to measure the frequency of observations.

B. Modal Class - Interval with the highest frequency (i.e. number of observations).

C. Frequency - Number of observations found for each interval. It is represented by the height of each bar in the graph.

D. Interval or Class - Set of real numbers between two values defined by exact limits. The shape of the histogram is influenced by the number and width of the intervals.

E. Interval width - Defined by the difference between the upper exact limit and the lower exact limit. The width is the same for all intervals (i.e. bars).

F. Horizontal axis - Scale of measure of the variable or CT characteristic.


When To Use A Histogram

  • Primarily used to display large amounts of data that are difficult to interpret in a spreadsheet format.
  • When you need to identify the centering, spread, and variation of the data in graphical form.
  • To quickly illustrate the underlying distribution of the data, usually a normal distribution.

The Histogram is commonly used to answer questions such as:

  • Is the process performing within the specification limits? 
  • Is there a wide variation in the process? 
  • If a change is required to the process, what change is appropriate? 

You can usually answer these questions by analyzing three key characteristics of the Histogram.

  1. How well is the data centered? The centering of the data provides information about the process target.
  2. How wide is the Histogram? The width of a Histogram defines its variability.
  3. What is the shape of the Histogram? If the shape is not a bell-shaped curve, which is usually the shape you are looking for, there is something going on in the process, which is causing quality problems.

Major Considerations

The measurement system must be validated prior to collecting data (when applicable). The guidelines to calculate the no. of intervals and their width must be observed.

Application Cookbook

1. Select the variable to measure (e.g. a CT characteristic).

2. Conduct a measurement validation study to ensure good measurements and repeatability (when applicable).

3. Collect and record data.

4. Define the number of intervals required to construct the graph. As a guideline the number of intervals is equal to (n) ½ where n represent the no. of observations.

  • Intervals should:
  • Be mutually exclusive
  • Be the same width
  • Be not less than 6 and not more than 20
  • Be continuous throughout the distribution
  • Limits of each interval are recorded in limit values
  • Limits of each interval are written in exact limits

5. Calculate the interval or class width using exact limits. Exact limits are extensions of plus or minus one-half the smallest unit that the measuring instrument can read. The smallest unit read is also known as resolution, so an exact limit is an extension of one half of the resolution.

6. Construct a frequency table counting how many observations fall into each interval when the limits are recorded using exact limits.

7. Create a bar chart using the count per interval.

ALTERNATIVE COOKBOOK USING MINITAB

8. Perform steps 1-4 as indicated above.

9. Collect and record data.

10. Enter the data in columns of the data window.

11. Choose the menu GRAPH.

12. Choose the sub-menu HISTOGRAM.

13. Select the column(s) containing the data in GRAPH VARIABLES and press OK.

Note: Minitab calculates the no. of intervals and the interval width.

Relative Frequency


Purpose

To depict the relative frequencies of numerical or measurement data expressed as percentage in a bar chart.

Anatomy

how to create a histogram

Reference: Juran's Quality Control Handbook

Terminology

A. Vertical axis - Scale to measure the frequency of observations expressed as percentage.

B. Percentage - Percent of the total observations found for each interval. It is represented by the height of the bar in the graph.

C. Interval - Set of real numbers between two values defined by exact limits.

D. Horizontal axis - Scale of measure of the variable or CT characteristic measured.


Major Considerations

The measurement system must be validated prior to collecting data (when applicable). The guidelines to calculate the no. of intervals and their width must be observed.

Application Cookbook

1. Perform steps 1-6 as indicated in the cookbook for "Raw Data".

2. Determine the total number of observations

3. Count how many observations fall into each interval of the frequency table.

4. Divide the value found in step 3 by the total from step 2 and express it as a percentage of the total number of observations. Record this value in the table.

5. Create a bar chart using the percentage found for each interval.

ALTERNATIVE COOKBOOK USING MINITAB

6. Perform steps 1-3 as indicated in the cookbook for the tool "Histogram, Raw".

7. Enter the data in columns of the data window.

8. Choose the menu GRAPH.

9. Choose the sub-menu HISTOGRAM.

10. Select the column(s) containing the data in GRAPH VARIABLES.

11. Select OPTIONS.

12. Select Type of Histogram: Percent.

13. Press on OK in the to menu boxes for MINITAB to create the graph.

Cumulative Relative Frequency


Purpose

To depict the cumulative relative frequencies of numerical or measurement data expressed as percentage in a bar chart.

Anatomy

how to create a histogram

Terminology

A. Vertical axis - Scale to measure the frequency of observations expressed as cumulative percentage

B. Cumulative Percentage - Cumulative percent of the total observations found for each interval. It is represented by the height of the bar in the graph

C. Interval - Set of real numbers between two values defined by exact limits

D. Horizontal axis - Scale of measure of the variable or CT characteristic measured

E. Cumulative histogram.

F. Cumulative frequency polygon constructed by connecting the midpoints of the intervals of the cumulative histogram (E)


Major Considerations

The measurement system must be validated prior to collecting data (when applicable). The guidelines to calculate the number of intervals and their width must be observed.

Application Cookbook

1. Perform steps 1-6 as indicated in the cookbook for "Raw Data".

2. Determine the total number of observations

3. Count how many observations fall into each interval of the frequency table.

4. Divide the value found in step 3 by the total from step 2 and express it as a percentage of the total number of observations. Record this value in the table.

5. Calculate the cumulative percent frequency for each interval.

6. Create a bar chart using the cumulative percentage found for each interval.

ALTERNATIVE COOKBOOK USING MINITAB

7. Perform steps 1-3 as indicated in the cookbook for the tool "Histogram, Raw".

8. Collect and record data.

9. Enter the data in columns of the data window.

10. Choose the menu GRAPH.

11. Choose the sub-menu HISTOGRAM.

12. Select the column(s) containing the data in GRAPH VARIABLES.

13. Select OPTIONS.

14. Select Type of Histogram: Cumulative Percent.

15. Press on OK in the to menu boxes for MINITAB to create the graph.

16. To create the cumulative percentage polygon select CONNECT in the menu DATA DISPLAY in MINITAB.

From Histogram to Six Sigma Tools.

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