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Unit 2: Normal Distributions

Alexis, Sommy , Elizabeth, Charlie. Unit 2: Normal Distributions. Density Curves. Always on or above the horizontal axis Has an area of exactly one underneath it Describes the overall pattern of the distribution and shows the proportion of all observations within a certain interval

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Unit 2: Normal Distributions

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  1. Alexis, Sommy, Elizabeth, Charlie Unit 2: Normal Distributions

  2. Density Curves • Always on or above the horizontal axis • Has an area of exactly one underneath it • Describes the overall pattern of the distribution and shows the proportion of all observations within a certain interval • Median: “equal areas point”, divides the area of the curve in half • Mean: “balance point”, the point at which the curve would balance if made of sold material

  3. Solving problems with density curves • Plot your data, make a graph, usually a histogram or stem plot • Look for a pattern (SOCS) • Calculate a numerical summary to describe center and spread

  4. Empirical Rule • In a normal distribution with mean µ and standard deviation σ • 68% of the observations fall within σ of the mean µ • 95% of the observations fall within 2σ of the mean µ • 99.7% of the observations fall within 3σ of the mean µ

  5. Percentile • Pth percentile is the value such that p percent of the observations fall at or below it • Often used for test scores in which the data is normally distributed

  6. Comparing Distributions • Z-scores • Percentiles: p percent of values that fall at or below the given number • Can use either z-scores or percentiles to compare data across two different distributions

  7. Standardizing Data • Z score: tells how many standard deviations away from the mean and the original observation falls and in which direction • Observations larger than the mean are positive when standardized • Observations smaller than the mean are negative when standardized

  8. Standard Normal Table • Gives the area under the standard normal curve • The table entry value z is the area under the curve left of z

  9. Normal Proportions Calculations • State the problem in terms of the observed variable x • Draw a picture of the distribution and shade the area of interest under the curve • Standardize x to restate the problem in terms of the normal variable z. Draw a picture to show the area of interest under the standard normal curve • Use the table to find the required area under the standard Normal curve • Write your conclusion in context

  10. Normal Probability Plot • Used to see if a normal model is adequate for the data • If the points lie close to a straight line the plot indicates that the data are Normal • Systematic deviations from a straight line indicate a Non-Normal distribution • Outliers appear as points that are far from the overall pattern of the plot

  11. Calculator Keystrokes • Histogram • STAT > 1. Edit > ENTER > L1>ENTER> (enter data) > 2nd Y = > 1. Plot > ENTER > On> ENTER> arrow over to histogram drawing> ENTER> arrow down to X list> L1 (second 1)> GRAPH • Z-score • 2nd VARS> 2. Normalcdf( > lower (lowest score) >ENTER> upper( maximum score) > ENTER> mean of data> ENTER > standard deviation of data > ENTER> paste> ENTER> answer is z score from table • Score needed • 2nd VARS> 3. invNorm( > area (z-score) >ENTER> mean of data> ENTER > standard deviation of data > ENTER> paste> ENTER> answer is score needed to lie in that area of distribution • Five Number Summary • STAT > 1. Edit > ENTER > L1>ENTER> (enter values) > STAT> arrow right to CALC> 1. 1-VAR Stats> ENTER> List (L1)> arrow down to calculate> ENTER • (mean, standard deviation, min, Q1, Median, Q3, Max)

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