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Normal Distribution

Normal Distribution. A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, etc.

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Normal Distribution

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    1. Normal Distribution Bell Curve

    2. Normal Distribution A normal distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, lengths of objects produced by machines, etc.   Certain data, when graphed as a histogram (data on the horizontal axis, amount of data on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.

    3. Normal Distribution A normal distribution is mathematically symmetrical right and left halves of the curve look exactly alike the mode, median and mean are identical and are located in the exact center of the of curve.

    5. A look at skewed distributions A distribution is skewed if one of its tails is longer than the other. The first distribution shown has a positive skew. This means that it has a long tail in the positive direction. The distribution below it has a negative skew since it has a long tail in the negative direction. Finally, the third distribution is symmetric and has no skew.

    6. Mean and Median This distribution has a positive skew. Note that the mean is larger than the median.

    7. Mean and Median This distribution has a negative skew. The median is larger than the mean.

    8. The outliers will pull the mean down the scale a great deal. The median might change due to the outlier The mode will be unaffected. Mean, Median and Mode

    9. Thus, with a negatively skewed distribution the mean is numerically lower than the median or mode.

    10. Thus, with a positively skewed distribution the mean is numerically higher than the median or the mode.

    11. Visual Mean and Median Measures of Central Tendency

    12. Mode, Median and Mean

    13. Mode

    14. Closer Look at Normal Distribution The spread of a normal distribution is controlled by the standard deviation The smaller the standard deviation the more concentrated the data Remember Standard Deviation is a measure of spread Bigger standard deviations mean the data is spread out more than lower standard deviations

    17. Percentages

    18. If you add percentages, you will see that approximately: 68% of the distribution lies within one standard deviation of the mean 95% of the distribution lies within two standard deviations of the mean. 99.8% of the distribution lies within three standard deviations of the mean.    It Adds Up

    19. Percentiles and the Normal Curve The mean (at the center peak of the curve) is the 50% percentile. The term "percentile rank" refers to the area (probability) to the left of the value. Adding the given percentages from the chart will let you find certain percentiles along the curve.

    22. Example 1 Find the percentage of the normally distributed data that lies within 2 standard deviations of the mean. Solution:  Read the percentages from the chart from -2 to +2 standard deviations. 95.4%

    23. Example 2 At the New Age Information Corporation, the ages of all new employees hired during the last 5 years are normally distributed.  Within this curve, 95.4% of the ages, centered about the mean, are between 24.6 and 37.4 years.  Find the mean age and the standard deviation of the data.

    24. Answer Solution:  As was seen in Example 1, 95.4% implies a span of 2 standard deviations from the mean.  The mean age is symmetrically located between -2 standard deviations (24.6) and +2 standard deviations (37.4).     

    25. Answer The mean age is 31 years of age. From 31 to 37.4 (a distance of 6.4 years) is 2 standard deviations.  Therefore, 1 standard deviation is (6.4)/2 = 3.2 years.

    26. The amount of time that Carlos plays video games in any given week is normally distributed.  If Carlos plays video games an average of 15 hours per week, with a standard deviation of 3 hours, what is the probability of Carlos playing video games between 15 and 18 hours a week?

    27. Solution:  The average (mean) is 15 hours.  If the standard deviation is 3, the interval between 15 and 18 hours is one standard deviation above the mean, which gives a probability of 34.1% or 0.341, as seen in the chart at the top of this page.

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