Statistics Unit
2.10 The Normal Distribution
The normal distribution is the most commonly-used probability distribution in all of statistics.
![A graph of a normal curve. The number line below the curve is numbered from -3 to 3 counting by 1. The graph is close to the number line on the left side of -3 and rises to a peak at 0 then goes back down close to the number line past 3.](images/u2s10intro1.png)
It has the following properties:
- Bell shaped: most data values clustered near the mean, well-defined single peak
- Symmetrical: data values spread evenly around the mean, larger deviations from the mean become increasingly rare, tapering tails of the distribution.
- Has one mode (peak)
- Mean and median are equal; both are located at the center of the distribution
Examples of things that are normally distributed.
- Birthweight of Babies
- ACT Scores
- Average NFL Player Retirement Age
![A histogram of the distribution of Newborn Weights. Frequency in thousands is labeled on the y axis and Weight in pounds is labeled on the x axis. The y-axis ranges from 0 at the bottom to 160 at the top in intervals of 20. The x-axis ranges from 6.7 to 8.5 in intervals of 0.1, with each bar of the histogram over one of the numbers. Each bar of the histogram gets higher from left to right until the middle of the graph when they go back down. The bars are symmetric on either side of the middle bar.](images/u2s10introa.png)
![A histogram of the distribution of ACT Scores. Frequency in thousands is labeled on the y axis and the score is labeled on the x axis. The y-axis ranges from 0 at the bottom to 160 at the top in intervals of 20. The x-axis ranges from 13 to 31 in intervals of 1, with each bar of the histogram over one of the numbers. Each bar of the histogram gets higher from left to right until the middle of the graph when they go back down. The bars are symmetric on either side of the middle bar.](images/u2s10introb.png)
![A histogram of the distribution of NFL Player Retirement Ages. Frequency is labeled on the y axis and the age is labeled on the x axis. The y-axis ranges from 0 at the bottom to 160 at the top in intervals of 20. The x-axis ranges from 25 to 43 in intervals of 1, with each bar of the histogram over one of the numbers. Each bar of the histogram gets higher from left to right until the middle of the graph when they go back down. The bars are symmetric on either side of the middle bar.](images/u2s10introb.png)
![There are three graphs in the picture. All three graphs are of the same bell-shaped curve, a normal distribution curve. The x-axis shows the mean in the middle, the highest point of the graph. Three standard deviations to the left and to the right are also marked on the graph. So, the x-axis reads, -3, -2, -1, mean, +1, +2, +3. The first graph is shaded below the curve between -1 and +1 to show that 68% of all data in a normal distribution will be between -1 and +1 standard deviations from the mean. The second graph is shaded below the curve from -2 to +2 to show that 95% of the data in a normal distribution is between -2 and +2 standard deviations from the mean. The third graph is shaded below the curve from -3 to +3 to show that 99.7% of the data in a nomral distribution is between -3 and +3 standard deviations from the mean.](images/u2s10intro3.png)
- The amount of Coca-Cola in a two-liter bottle is normally distributed with a mean of 2.05 liters and a standard deviation of 0.1 liters. (0.1 liters is approximately 3 ounces or 0.4 cups)
Sketch a distribution curve and label.
- What percentage of two-liter Coca-Cola bottles contain between 1.95 and 2.15 liters?
- What percentage of two-liter Coca-Cola bottles contain more than 2.05 liters?
- The range of liters that approximately 95% of all Coca-Cola bottles is from ___________ to ___________.
- What percentage of two-liter bottles contain between 1.85 and 2.15 liters?
- What percentage of two-liter bottles contain less than 1.85 liters?
- In a sample of 10,000 two-liter Coca-Cola bottles, how many would be expected to contain more than 2.35 liters? (This is almost 1.5 cups above the advertised 2 liters)
68%
50%
1.85 to 2.25 liters
81.5%
2.5%
\(.0015 \cdot 10000 = 15\) bottles
- The average gas mileage for a hybrid Honda Civic is normally distributed with a mean of 44 miles per gallon and a standard deviation of 5 mpg.
Sketch a distribution curve and label.
- What percentage of Honda Civic hybrids get better than an average of 49 miles per gallon?
- What percentage of Civic hybrids get worse than an average of 39 miles per gallon?
- What percentage of Civic hybrids get between 29 and 39 miles per gallon?
- The range in miles per gallon for 99.7% of Civic Hybrids is from ______________ to ______________.
- In a sample of 500 Civic hybrids, how many would be expected to get between 39 and 49 miles per gallon on average?
- In a sample of 500 Civic hybrids, how many would be expected to get better than an average of 59 miles per gallon?
16%
16%
15.85%
29 to 59 mpg
\(0.68 \cdot 500 = 340\) Honda Civic Hybrids
\(0.0015 \cdot 500 = 0.75\)
Less than one Honda Civic Hybrid per 500 Civics will get between 39 and 49 mpg.