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Standard Normal Distribution Table
What is a standard normal distribution table? What statistics are used with a standard normal distribution? Can other distributions be converted to a standard normal distribution?
Types of Frequency Distribution Charts
A frequency distribution chart uses a graph to show the incidence of a particular trait. The X axis indicates a variable or value, such as household income or the width of a manufactured part. The Y axis indicates the number of occurrences for the variable. Curves can be negatively skewed or skewed to the left, positively skewed or skewed to the right or bell-shaped.
Some frequency distributions, such as household income, are naturally skewed. Other variables, such as human height, are more symmetrical and evenly distributed. The bell-shaped frequency distribution is called the normal distribution. In a normal distribution, the most common frequency is in the middle and each side is symmetrical.
What If My Distribution Table Has More Than One Peak?
Bi-modal distributions have two peaks. Multi-modal distributions have three or more peaks on the distribution table. If your frequency distribution has two or more peaks, you may have a mixture of two different normal distributions. Input from two different shifts or suppliers create more than one peak; separate the data by source to get the normal distribution table for each sample.
The true curves of a standard normal distribution may become clearer after you have far more data points to gain a clear and much smoother distribution table. If the data still generates more than one peak, you cannot use a standard normal distribution table and related statistics to determine the odds of a value occurring or what percentage of product will fall outside of a selected range such as specification limits.
Statistics Derived from Normal Distributions
In a standard normal distribution, the mean, the median and the mode all occur at the same point, the peak of the bell-shaped curve. Standard deviation is the positive square root of the variance.
For a standard normal distribution, the interval plus and minus one standard deviation from the middle of the mean will include roughly 68% of all occurrences.The range plus and minus two standard deviations from the mean will include roughly 95% of all data points. Plus or minus three standard deviations will include 99.7% of all data points.
The term Six Sigma refers to plus and minus six standard deviations from the mean on a standard normal distribution, and this range will include all values except 37 out of a million data points.
Standard Normal Curves and Probabilities
The probability of a particular occurrence, variable value or range of values can be calculated using the standard normal curve. To determine the probability of an occurrence, the area under the normal curve is set to one. The probability of a value or range of values is now found based upon the area of that section of the standard normal distribution. The area between the mean and one standard deviation on either side of the mean is 0.3413.
The odds of the variable falling within the median and one standard deviation is 34.13%. Converting the variables to a Z-score simplifies the process of determining the probability of a particular value or range of values.
A Z-score is calculated using the equation X – (mean) divided by the standard deviation. The Z-score can then be referenced on a standard normal table to find the probability of that value occurring. The Z-score will tell you the odds of that value or a lower one being found. Z-scores can be positive or negative or even equal zero.
If the mean of a data set is 20 and the standard deviation is 10, let’s calculate the Z-score. For a value of 30, the Z-score is (30-20)/10 or 1.0. For a value of 10, the Z-score is (10-20)/10 or -1.0.
The Z-score allows you to determine the probability of your product and its variation falling outside of the control limits that you set or find the statistical control limits that will contain 90% of your manufactured product. You can use a Z-score to determine how many outliers or exceptions you will see if you set the cut off at a specific level, such as how many people will not have their issue resolved if customer support staff can only spend five minutes on the phone with each. Conversely, the Z-score will tell you at what point 95% of all calls would be resolved, based on your particular standard normal curve.
The Z-score can handle data from any source and any process, converting it to a number that is statistically valid as long as the underlying process is not changing and generates a standard normal distribution. Z-scores can also be used to determine confidence intervals. A greater level of confidence will increase the width of the confidence interval.
Control Limits and Standard Normal Distributions
Statistical process control limits use statistics to set control limits, the lines on a run chart that indicate when a product is out of spec. While a run chart will bounce between the control limits most of the time, it will occasionally generate a data point outside of the specification limits. If all of these data points were put in a frequency distribution chart, you will see a standard normal distribution.
The middle line of the control chart is the median or mean of the standard normal distribution. A run chart with a series of points near the control limit or an increasing number of out of spec data points means that the average or process variability is changing.
Approximating Standard Normal Distributions with Binomial Distributions
Most other types of distributions cannot be converted to standard normal distributions to take advantage of the Z score and other simplified statistics that work with standard normal distributions. Binomial distributions can approximate a normal distribution.
The mean for a binomial distribution is approximated by multiplying the number of trials by the odds of success for a single trial. The standard deviation is the square root of the number of trials multiplied by the total number of successes divided by the number of trials. Using the normal approximation of a binomial distribution allows you to use Z scores to estimate the probability of a particular outcome or set of outcomes.
1. "Statistics for Business and Economics" by David Ray Anderson et al.
2. "The Complete Idiots Guide to Lean Six Sigma" by Neil DeCarlo