Chapter 6 The Standard Deviation as a Ruler and the Normal Model

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Chapter 6The Standard Deviation as a Ruler and
the Normal Model

• Math2200

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Examples How do we compare measurements on
different scales ?

• SAT score 1500 versus ACT score 21
• Womens heptathlon
• 200-m runs, 800-m runs, 100-m high hurdles
• Shot put, javelin, high jump, long jump
• Make them on the same scale by standardization
• Count how many standard deviations away from the
  mean

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The Standard Deviation as a Ruler

• The standard deviation tells us how the whole
  collection of values varies, so its a natural
  ruler for comparing an individual to a group.
• Standardizing with z-scores
• We compare individual data values to their mean,
  relative to their standard deviation using the
  following formula
• We call the resulting values standardized values,
  denoted as z. They can also be called z-scores.

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Heptathlon Kluft versus Skujyte
Carolina Kluft (Sweden) Gold Medal
Austra Skujyte (Lithuania) Silver Medal
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Kluft versus Skujyte
Long jump shot put
mean 6.16m 13.29m
sd 0.23m 1.24m

Kluft 6.78m 14.77m
z-score 2.70 (6.78-6.16)/0.23 1.19 (14.77-13.29)/1.24

Skujyte 6.30m 16.40m
z-score 0.61 (6.30-6.16)/0.23 2.51(16.40-13.29)/1.24

Total z-score
Kluft 2.701.193.89
Skujyte 0.612.513.12
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Standardizing with z-scores (cont.)

• Standardized values (z-scores) have no units.
• z-scores measure the distance of each data value
  from the mean in standard deviations.
• A negative z-score tells us that the data value
  is below the mean, while a positive z-score tells
  us that the data value is above the mean.

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Benefits of Standardizing

• Standardized values have been converted from
  their original units to the standard statistical
  unit of standard deviations from the mean.
  Thus we can compare values that are measured on
  different scales, or in different units.

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Shifting Data

• Adding (or subtracting) a constant to every data
  value adds (or subtracts) the same constant to
  measures of position.

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Shifting the data

• Adding (or subtracting) a constant to each value
  will increase (or decrease) measures of position
  center, percentiles, max or min by the same
  constant. Its shape and spread, range, IQR,
  standard deviation - remain unchanged.

Mean c sd unchanged
Median c IQR unchanged
Min c
Max c
Q1 c
Q3 c
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Rescaling Data

• Multiply (or divide) all the data values by any
  constant

• The mens weight data set measured weights in
  kilograms. If we want to think about these
  weights in pounds, we would rescale the data

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Rescaling Data (cont.)

• All measures of position (such as the mean,
  median, and percentiles) and measures of spread
  (such as the range, the IQR, and the standard
  deviation) are multiplied (or divided) by that
  same constant.

Mean a sd a
Median a IQR a
Min a
Max a
Q1 a
Q3 a
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How does the standardization changes the
distribution?

• Standardizing data into z-scores shifts the data
  by subtracting the mean and rescales the values
  by dividing by their standard deviation.
• Shape
• No change
• Mean
• 0 after standardization
• Standard deviation
• 1 after standardization

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How should we read a z-score?
Long jump shot put
mean 6.16m 13.29m
sd 0.23m 1.24m

Kluft 6.78m 14.77m
z-score 2.70 (6.78-6.16)/0.23 1.19 (14.77-13.29)/1.24

Skujyte 6.30m 16.40m
z-score 0.61 (6.30-6.16)/0.23 2.51(16.40-13.29)/1.24

Total z-score
Kluft 2.701.193.89
Skujyte 0.612.513.12

• The larger a z-score is (negative or positive),
  the more unusual it is.
• How do we evaluate how unusual it is?

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Normal models

• Quantitative variables
• Unimodal symmetric
• N(µ,s)
• µ mean s sd
• Standard normal N(0,1)
• Parameters (µ,s)
• Statistics ( , )

• If (µ,s) are given,

• If (µ,s) are not given,

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Data, Model

• Data , Statistics
• Model , Parameter
• A tool to describe the data with a number of
  parameters
• Questions related to the data can be answered by
  the value of the parameters
• Estimate parameters by certain statistics
• Mean from the model, sample mean
• SD from the model, sample standard deviation
• Proportion parameter, sample proportion

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When to use the Normal model?

• When we use the Normal model, we are assuming the
  distribution is Normal.
• Nearly Normal Condition The shape of the datas
  distribution is unimodal and symmetric.
• This condition can be checked with a histogram or
  a Normal probability plot.

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68-95-99.7 rule

• How do we measure how extreme a value is using
  normal models?
• 68 within
• 95 within
• 99.7 within

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Finding normal probability using TI-83

• 2nd VARS (DISTR)
• normalcdf(lowerbound, upperbound, µ,s)
• If the data is from N(0,1), what is the chance to
  see a value between -0.5 and 1?
• normalcdf(-0.5,1,0, 1) 0.5328072082
• If the data follow the N(1,1.5) distribution,
  what is the chance to see a value less or equal
  to 5?
• normalcdf(-1E99, 5,1,1.5) 0.9961695749

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From probability to Z-scores

• For a given probability, how do we find the
  corresponding z-score or the original data value.
• Example What is Q1 in a standard Normal model?
• TI-83 invNorm(probability, µ,s)
• invNorm(0.25,0,1) -0.6744897495

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Example SAT score

• A college only admits people with SAT scores
  among the top 10. Assume that SAT scores follow
  the N(500,100) distribution. If you want to be
  admitted, how high your SAT score needs to be?
• invNorm(0.9,500,100) 628.1551567

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Finding the parameters

• While only 5 of babies have learned to walk by
  the age of 10 months, 75 are walking by 13
  months of age. If the age at which babies develop
  the ability to walk can be described by a Normal
  model, find the parameters?
• Z-score corresponding to 5
• (10- µ)/ s invNorm(0.05,0,1) -1.644853626
• Z-score corresponding to 75
• (13- µ)/ s invNorm(0.75,0,1) 0.6744897495
• Solve the two equations, we have
• µ12.12756806
• s1.293469536

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Normal Probability Plots

• If the distribution of the data is roughly
  Normal, the Normal probability plot approximates
  a diagonal straight line. Deviations from a
  straight line indicate that the distribution is
  not Normal.
• Nearly Normal data have a histogram and a Normal
  probability plot that look somewhat like this
  example

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Normal Probability Plots (cont.)

• A skewed distribution might have a histogram and
  Normal probability plot like this

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How to make Normal Probability Plots?

• Suppose that we measured the fuel efficiency of a
  car 100 times
• The smallest has a z-score of -3.16
• If the data are normally distributed, the model
  tells us that we should expect the smallest
  z-score in a batch of 100 is -2.58.
• This calculation is beyond the scope of this
  class.
• Plot the point whose X-axis is for the z-scores
  given by the normal model and Y-axis is for that
  from the data
• Keep doing this for every value in the data set
• If the data are normally distributed, the two
  scores should be close, and graphically, all the
  points should be roughly on a diagonal line.

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What Can Go Wrong?

• Dont use a Normal model when the distribution is
  not unimodal and symmetric.

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What Can Go Wrong? (cont.)

• Dont use the sample mean and sample standard
  deviation when outliers are presentthe mean and
  standard deviation can both be distorted by
  outliers.
• Dont round your results in the middle of a
  calculation.