Z-score and probability in statistics.pdf

Variance
• Variance is a measurement of the spread between numbers in a data
set.

Standard Deviation
• In statistics, the standard deviation is a measure of the amount of
variation of a random variable expected about its mean.
• A low standard deviation indicates that the values tend to be close to
the mean of the set, while a high standard deviation indicates that
the values are spread out over a wider range

Raw Scores
• The definition of a raw score in statistics is an unaltered
measurement.
• Raw scores have not been weighted, manipulated, calculated,
transformed, or converted. An entire data set that has been unaltered
is a raw data set.

Z - Score
• Z-score is a statistical measure that quantifies the distance between a
data point and the mean of a dataset.
• It's expressed in terms of standard deviations. It indicates how many
standard deviations a data point is from the mean of the distribution.

Z - Score
• For a recent final exam in STAT 500, the mean was 68.55 with
a standard deviation of 15.45.
• If you scored an 80%: 𝑍=(80−68.55)/15.45=0.74, which
means your score of 80 was 0.74 SD above the mean.
• If you scored a 60%: 𝑍=(60−68.55)/15.45=−0.55, which
means your score of 60 was 0.55 SD below the mean.

Z - Score
• The scores can be positive or negative.
• For data that is symmetric (i.e. bell-shaped) or nearly symmetric, a
common application of Z-scores for identifying potential outliers is for
any Z-scores that are beyond ± 3.

Using z-scores to standardise a distribution
• Every X value in a distribution can be transformed into a
corresponding z-score
• Any normal distribution can be standardized by converting its values
into z scores.
• Z scores tell you how many standard deviations from the mean each
value lies.
• Converting a normal distribution into a z-distribution allows you to
calculate the probability of certain values occurring and to compare
different data sets

Using z-scores to make comparison
• we can compare performance [values] in two different distributions,
based on their z-scores.
• Lower z-score means closer to the meanwhile higher means more far
away.
• Positive means to the right of the mean or greater while negative
means lower or smaller than the mean

• Jared scored a 92 on a test with a mean of 88 and a standard
deviation of 2.7. Jasper scored an 86 on a test with a mean
of 82 and a standard deviation of 1.8. Find the Z-scores for
Jared's and Jasper's test scores, and use them to determine
who did better on their test relative to their class.

• Step 1: Compute each test score's Z-score using the mean
and standard deviation for that test.
• For Jared's test, the Z-score is:
𝑍=(𝑥−𝜇)/𝜎 = (92−88)/2.7=4/2.7 = 1.48
• For Jasper's test, the Z-score is:
𝑍=(𝑥−𝜇)/𝜎 = (86−82)/1.8 = 4/1.8 = 2.22

• Step 2: Use Z-scores to compare across data sets.
• Jared's Z-score of 1.48 says that his score of 92 was between
1 and 2 standard deviations above the mean. Jasper's Z-score
of 2.22 says that his score of 86 was a bit more than 2
standard deviations above the mean. So, Jasper's score of 86
was relatively higher for his class than Jared's 92 was for his
class.

Probability
• Probability is simply how likely something is to happen.
• Whenever we're unsure about the outcome of an event, we
can talk about the probabilities of certain outcomes—how
likely they are.
• The analysis of events governed by probability is called
statistics.

What are Equally Likely Events?
• When the events have the same theoretical probability of happening, then
they are called equally likely events. The results of a sample space are
called equally likely if all of them have the same probability of occurring.
For example, if you throw a die, then the probability of getting 1 is 1/6.
Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one
at a time is 1/6. Hence, the following are some examples of equally likely
events when throwing a die:
• Getting 3 and 5 on throwing a die
• Getting an even number and an odd number on a die
• Getting 1, 2 or 3 on rolling a die
are equally likely events, since the probabilities of each event are equal

Random sampling
Simple random sample
• Each member of the population has an equal chance of being
selected
Independent random sample
• Each member of the population has an equal chance of being
selected
AND
• The probability of being selected stays constant from one selection
to the next [if more than one individual is selected]
• i.e. Sampling with replacement

Independent Random Sampling
• Probability of event A =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑐𝑙𝑎𝑠𝑠𝑖𝑓𝑖𝑒𝑑 𝑎𝑠 𝐴
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

Probability and Frequency distributions
• Probability usually involves a population of scores displayed in a
frequency distribution graph.
• What is the probability of obtaining an individual score of less than 3?
[i.e. either 1 or 2?]
N = 20

Probability and the normal distribution
• In any normal distribution the percentage of values that lie within a
specified number of standard deviations from the mean is the same

Graphing Probability …
68 – 95 -99.7% Rule of Thumb revisited
• One standard deviation either side of the mean captures:
• Approx 68% of our data
• Mathematically: 68.26%
• Two standard deviations either side of the mean captures:
• Approx 95% of our data
• Three standard deviations either side of the mean captures:
• Approx 99.7% of our data

68% – 95% -99.7% Rule of Thumb revisited
68.26% – 95.44% – 99.73% Maths calculation

Probability
What is the probability that a randomly selected data value in a normal distribution
lies more than 1 standard deviation below the mean?
p(z < - 1.00)
What is the probability that a randomly selected data value in a normal distribution
lies more than 1 standard deviation above the mean?
p(z > 1.00)

Calculating probability in a normal distribution
• When calculating the probability we should calculate the Z-Score
Standardise the distribution [z-score calculation],
z =
𝑋−𝜇
𝜎
If scores on a test were normally distributed with:
• mean of 𝜇 = 60, and a standard deviation of 𝜎 = 12,
• what is the probability [of a randomly selected person who took the
test] of a score greater than 84?

Z-score and probability in statistics.pdf

Probability using Unit Normal Table
• Quite often the values we are interested in are not exactly 1, 2 or 3
standard deviations away from the mean. Statistical tables [or online
probability calculators] can be used to calculate the probability

The body always corresponds to the larger part of the distribution
• can be located on the left or the right of the distributions
The tail always corresponds to the smaller part of the distribution
• again, can be located on the left or the right of the distributions

Example
Information from the department of Motor Vehicles indicates that the
average age of licensed drivers is 𝜇 = 45.7 years with a standard
deviation of 𝜎 =12.5 years. Assuming that the distribution of drivers’
ages is approximately normal,
1. What proportion of licensed drivers are older than 50 years old?
z =
𝑋−𝜇
𝜎
=
50−45.7
12.5
=
4.3
12.5
= 0.34
2. What proportion of licensed drivers are younger than 30 years old?
z =
𝑋−𝜇
𝜎
=
30−45.7
12.5
=
−15.7
12.5
= -1.26 [so, 30 is 1.26 sds below]

Examples
• The length of a human pregnancy is normally distributed with a mean
of 272 days with a standard deviation of 9 days .
1. State the random variable.
2. Find the probability of a pregnancy lasting more than 280 days.
3. Find the probability of a pregnancy lasting less than 250 days.
4. Find the probability that a pregnancy lasts between 265 and 280
days..
5. Suppose you meet a woman who says that she was pregnant for
less than 250 days. Would this be unusual and what might you
think?

Z-score and probability in statistics.pdf

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