Comparing normal distributions

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Comparing normal distributions
To compare two sets of data we use z-scores.
Using z-scores allows us to compare the relative
positions of two scores that are in different
distributions, that may have different means and
standard deviations. When we change our raw
scores into standardised scores, or z-scores, the
problems of different means and standard
deviations are overcome. This is because each
score is now expressed in relative values. The
way we use the z-score will depend upon the
question asked. If the question wants the best
result, we will use the biggest z-score. If the
question wants the worst result, we will use the
smallest z-score. If we want which score is
closest to the average of its distribution, we
will use the z-score closest to zero.
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Example 1
Emma compares her results in French and Japanese.
In French Emma received 82, the class average
was 68 and the standard deviation of the class
was 8.45. In Japanese Emma received 37 out of
50, the class average was 28 and the standard
deviation of the class was 5.3. In which subject
did Emma do best?

The subject Emma did best in was Japanese, as her
z-score was highest for Japanese.
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Example 2
Sam, a zoologist, is studying the weights of mice
and elephants. The breed of elephants she is
studying has a mean weight of 20.6 tonnes and a
standard deviation of 3.2 tonnes. The breed of
mouse Sam is studying has a mean weight of 35.6
grams and a standard deviation of 2.8 grams.
Which animal is more underweight, Dumbo the
elephant weighing 17.1 tonnes, or Mickey the
mouse weighing 31.4 grams?

Mickey the mouse is the more underweight, as his
z-score was lower than Dumbos.
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Example 3
Sinead and Paul compete in synchronised swimming
teams in different states. Sineads team, the
lovely legs, competes in Victoria, where they say
the judging is the hardest in the country. The
lovely legs scored 7.3 last weekend in a
competition that averaged 6.78 and had a standard
deviation of 0.42. Pauls team, the fabulous
floaters, competes in Queensland. The fabulous
floaters scored 8.4 two weeks ago in a
competition that averaged 7.43 and had a standard
deviation of 0.61. (a) Calculate the score the
lovely legs would have got if they competed in
the Queensland competition, assuming they
performed to the same level. (b) In which state
are the judges more consistent with their scores?
Calculate the z-score for the lovely legs.
Calculate the lovely legs score in the Queensland
competition.
b) Victoria had a standard deviation of 042. Qld
had a standard deviation of 061. ? Victoria had
the most consistent judges.
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Todays work
Exercise 8A pg 239 6, 7, 9, 10
Yesterdays work
Exercise 10-02 Q1?5 10