Youth Forward Scholarship Winter 2018 – My Service To My Community

Name: Lesslie Hasman
From: El Cajon, California
Grade: Junior
School: Santana High School
Votes: 0

Measures of Relative Standing and Density; Z-scores

  • _________________________measure a score as related to the other
    scores in the data set (Measures the relative standing).

  • _____________________________________________________(Z-score)

    • Puts everything on the same scale in terms of standard deviation.

    • Standardized values (z-score):

      • z =

        • Tells us how many ___________________________________________ the
          original observation is from the mean, and in which direction:

        • Observations larger than the mean are _________________________.

        • Observations smaller than the mean are

  • My husband and I argue about who is smarter….

    • My SAT score: 1350

      • N(1000, 200)

    • His ACT score: 28

      • N(18, 6)

  • Standardizing Women’s Heights: N(64.5, 2.5)

  1. A women 68 inches tall has a standardized height of what?

  1. A women 58 inches tall has a standardized height of what?

Measures of Relative Standing and Density Curves; Percentiles

  • We can standardize data by using z-scores, but we can also use


  • We say that the pth percentile of a distribution is the value
    where _________________ of the observations are less than or
    equal to it.

    • If I scored a 78 on my math test, which was better than 25 of my 28
      class mates, then I am in the 25/28=89 percentile for the class.

  • Example: Here are the scores of 25 math students in my class


























  • If your score was 80 on the test, what percentile would you be in on
    this test?

  • What about if your score was 72?

  • What about if your score was 89?

  • Have you wondered if there is an easy way to convert Z-scores to

  • The percent of observations falling at or below a particular Z-score
    depends on the________________________ of the distribution.

    • An observation that has a score equal to the mean has a Z-score of
      0. In a heavily left skewed distribution where the mean is less
      than the median, this observation will be below the 50th
      percentile (the median).

2.1C Measures of Relative
Standing and Density Curves; Density Curves

Density Curves

  • Sometimes the overall pattern of a distribution is such that we can
    describe it with a __________________________________. It is
    remarkable how many natural phenomena appear to be related to

__________________________________________________. When appropriate,
using a normal distribution model to represent distributions that
occur in real-life situations can be extremely useful in statistical

  • __________________________________________ – a mathematical model
    for the distribution

    • Displays the overall pattern (____________________) of a
      distribution, but is an idealized description, ignoring minor

    • Has an area of exactly
      _________________________________ underneath it.

    • Is on or above the

    • The area under the curve and above any range of values is the
      proportion of all observations that fall in that range.

    • Types

      • Normal curve (symmetric normal density curve)

      • Right-skewed curve

      • Left-skewed curve

The median and mean of a
density curve

  • ____________________________________________________________

    • The point that divides the area under the curve into

    • _________________________________________________: half the area
      under the curve to its left and the remaining half of the area to
      its right.

  • ____________________________________________________________

    • The “balance point” of the curve (think of a

    • The point at which the curve would balance if made of solid

  • ____________________________________________________________________________________________

    • The median and mean are the same for a symmetric density curve.
      They both lie at the ____________________________ of the curve.

    • The mean of a skewed curve is pulled _________________________ from
      the median in the direction of the

Normal Distributions

Normal Distributions

  • ________________________________________

    • Density curves that are _________________________,
      ______________________________, and

    • Describe normal distributions.

    • Described by its ___________________
      and ________________________________________

      • is
        located at the ____________________ of the symmetric curve and is
        the _____________________ as median.

      • controls
        the __________________________ of a normal curve.

        • A curve with a larger standard deviation is

  • Locating
    by eye:

  1. Imagine you are skiing down a mountain that has the shape of a

  2. At first the descent is _______________________________.

  3. Then the slope begins to grow ________________________ as you go out
    and down.

  4. You can feel the change as you run a pencil down a normal curve.

  • The points at which this change of curvature takes place are
    __________________ _________________________
    and are located at distance

    on either side of the mean .

  • Why are the normal distributions important in statistics?

  1. Normal distributions are good descriptions for some distributions of

(i.e. SAT exam scores, psychological test scores, lengths of

  1. Normal distributions are good ____________________________ to the
    results of many kinds of __________________________________________
    (i.e. tossing a coin many times).

  2. Many _______________________________________ procedures based
    on normal distributions work well for other
    __________________________________ distributions.

_________________________________ rule

  • In a normal distribution with mean
    and standard deviation

    • _____________
      of the observations fall within __________ of

    • _____________ of the observations fall within __________ of

    • _____________ of the observations fall within ___________

  • A short notation for the normal distribution
    with mean

    and standard deviation

    • N(,)

    • For example: N(64.5, 2.5)

Normal Distribution; Standard Normal Distributions

Normal Distribution:

standard normal distribution is the normal distribution with
______________ where ________________and ________________.

  • z = Standardized
    Normal Distribution

Distribution Calculations:

  • An area under a density curve is

Any question about what proportion of observations lie in some range
of values can be answered by finding an area under the curve.

  • Table A is a table of areas under the standard normal curve.
    The table entry for each value z is the area under the curve
    to the left of z.

z-score chart (go to closest #.
vs. <? no difference) – Draw sketches

  1. z -2.25

  1. z -2.25

  1. z > 1.77

  1. –2.25 z

  • I wonder what percentile I lie in for my SAT scores:

    • 68-95-99.7 rule will only give percentiles for 1, 2, or 3
      standard deviations from the mean, but I am 1.15 standard
      deviations above the mean.

  • Finding normal proportions

Step 1: State
the problem in terms of the observed variable

_____________________ of the distribution and
_________________ the area of interest under the curve.

Step 2:
______________________________ to restate the problem in terms of a
standard normal variable
Draw a picture to show the area of interest under the standard
normal curve.

Step 3: Find the area under the standard normal curve, using
_____________________ and the fact that the total area under the
curve is __________.

Step 4: Write your conclusion in the context of the problem.

  • So, what is the area below 1.75 standard deviations?

  • More Practice

1) What percent of people score less than 800 on the SAT if
N(1000, 200)?

2) What percent of people score more than 1400 on the SAT?

3) What percent of people score between 900 & 1350 on the

Normal Distributions; Standard Normal Curve and Assessing Normality

Finding a value given a

  • We may want to find the observed value with a given proportion of
    the observations above or below it. To do this,

      • Find the given proportion in the _________________ of the table.

      • Read the corresponding z from the left column and top row.

      • “_____________________________________” to get the observed

  • Scores on the SAT verbal test in recent years follow approximately
    the N(505, 110) distribution. How high must a student score
    in order to place in the top 10% of all students taking the SAT?

  1. ____________________________________________. We want to find
    the SAT score x with are 0.1 to its right under the
    normal curve with mean = 505 and
    standard deviation = 110. That’s
    the same as finding the SAT score x with area 0.9 to its

  2. _____________________________________. Look in the body of
    Table A for the entry closest to 0.9. It is 0.8997. This is the
    entry corresponding to z = 1.28. So z = 1.28 is the
    standardized value with area 0.9 to its left.

  3. ______________________________________ to transform the
    solution from the z back to the original x scale.

Solving this equation for x gives:

This equation should make sense: it says that x lies
___________________ above the mean on this particular normal curve.
That is the “unstandardized” meaning of z

A student must score at least _________________ on the SAT verbal
test to place in the highest 10% of students who take the test.

  • The general formula for unstandardizing a z-score:

  • N(1000, 200) What score do I need to be in the top 5% so I
    can get into Harvard?

would need to score at least ______________________ to place in the
top 5% of students who take the SAT.


  • In the latter part of this course we will want to invoke various
    tests of significance to try to answer questions that are important
    to us.

      • These tests involve sampling people or objects and inspecting them
        carefully to gain insights into the populations from which they

      • Many of these procedures are based on the assumption that the host
        population is approximately normally distributed.

      • Consequently, we need to develop methods for assessing normality.

Construct a ________________________________________________
or a ______________________________. See if the graph is
approximately normal about the mean.

Use the __________________________________ to check for

Construct a
A normal probability plot provides a good assessment of the adequacy
of the normal model for a set of data. Use statistics utilities like
Minitab, or a TI-83 or TI-84 calculator to construct normal
probability plots.

the data distribution is close to a normal distribution, the plotted
points will lie close to a ___________________________________.
Conversely, nonnormal data will show a
________________________________ trend. Outliers appear as points
that are far away from the overall pattern of the plot.

normal distribution produces a
________________________________________ on the plot because
standardizing is a transformation that can change the slope and
intercept of the line in our plot but cannot change a line into a
curved pattern.

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