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Section 2.5 The Normal Distribution. Statistics Chapter 2 Exploring Distributions. Central Intervals for Normal Dist. 68% of values lie within 1 SD of the mean. Including to the right and left 90% of the values lie with 1.645 SDs of the mean.

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Statistics Chapter 2 Exploring Distributions

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Section 2.5

The Normal Distribution

StatisticsChapter 2 Exploring Distributions

Central Intervals for Normal Dist.

  • 68% of values lie within 1 SD of the mean.

    • Including to the right and left

  • 90% of the values lie with 1.645 SDs of the mean.

  • 95% lie within about 2 SDs (actually 1.96 SDs) of the mean.

  • 99.7% of the data lie within 3 SDs of the mean.

Why do we study the Normal Distribution?

  • Very common distribution of data throughout many disciplines.

    • SAT / ACT scores

    • Measure of diameter of tennis balls

    • Heights / weights of people

  • Once we know a distribution is Normal, there is a tremendous amount of information we can determine or predict about it.

Normal Distribution

  • All normal distributions have the same basic shape.

    • The difference: tall and thin vs. short and fat

    • However, we could easily stretch the scale of the tall thin curve to make it identical to the short fat one.

  • The area under the curve can be thought of in terms of proportions or percentage of data.

    • The total area under the curve is 1.0 (100%)

The Standard Normal Distribution

  • We can standardize any normal curve to be identical.

  • We do this by treating the mean as Zero and the SD as One.

  • The variable along the x-axis becomes what we call a z score.

  • The z score is the number of SDs away from the mean.

Finding z scores for the Standard Normal Distribution

  • Practice problems:

  • 1) Normal distribution with:

    • mean = 45 and SD = 5

    • Find the z score for a data value of 19

    • Find the z score for a data value of 52

  • 2) Normal distribution with:

    • Mean = 212 and SD = 24

    • Find the z score for a data value of 236

Proportion of data in a range

  • We can use the standard normal curve to find proportion of data in a range of values.

  • Normal Curve example: SAT I Math scores

    • Mean = 500 SD = 40

    • Find the proportion of data in the score range 575 or less.

  • Using z tables: Table A very back of book

    • Find the proportion of data above 575.

    • Find the proportion of data between 490 and 550.


  • Read all of 2.5.

  • Be prepared for quiz on Tuesday.

2.5 Quiz

  • You have collected data regarding the weights of boys in a local middle school. The distribution is roughly normal. The mean is 113 lbs and the SD is 10 lbs.

    • A) What proportion of boys are below 100 lbs?

    • B) What proportion are above 120 lbs?

    • C) What proportion are in between 90 & 120 lbs?

Using calculator for proportions

  • You can also use the TI-83 or higher to find these same proportions:

  • 2nd , Distr, normalcdf(low, high,mean,SD)

  • When using z scores you can leave mean,SD blank. normalcdf(low, high) it will default to mean=0 and SD=1.

  • This will give you the same area under the curve (proportion of data) as the z table.

Finding the z score from the Percent

  • If you know the percent of data covered under a normal distribution, you can find the z-score.

  • Simply look up the percent (proportion) in the z table and relate it to the corresponding z score.

    • Find the value that is closest to the percent given

  • Another method is with the calculator.

    • 2nd ,Distr, invNorm(proportion, mean, SD)


  • Find the z-score that has the given percent of values below it in a standard normal distribution:

    • a) 32% b) 41% (use the z-table)

    • c) 87% d) 94%(use your calculator)

Using the z-score to find a value

  • If you know how many SDs a value is from the mean, you can use this (z-score) to find the actual data value:

    • x = mean + (z • SD)

  • Example: The mean weight of the boys at a middle school is 113 lbs, with a SD of 10 lbs. One boy is determined to be 2.2 SDs above the mean. How much did the boy weigh?

Combining the last two situations

  • So now, if you know the percentage of data above or below a data value and you know the mean and SD, you can figure out that data value:

    • Use z-table to find the z-score, then use the z score with mean and SD to find the data value.

    • Or you can use the invNorm function on your calc.

      • invNorm(proportion, mean, SD)


  • The heights of U.S. 18-24 yr old females is roughly normally distributed with a mean of 64.8 in. and a SD of 2.5 in.

    • Estimate the percent of women above 5’8”

    • What height would a US female be if she was 1.5 SDs below the mean? Give your answer in ft & in.

    • What height would a US female be if she was considered to be in the 80th percentile?

Review Examples

  • What percentage of US females is above 5’7”?

  • What percent are between 5’7” and 5’0”?

Trick Question

  • The cars in Clunkerville have a mean age of 12 years and a SD of 8 years. What percentage of cars are more than 4 years old?

    • Why is this a trick question?


  • Page 93

    • E59, 61, 63, 64, 67, 69, 71, 73, 74

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