Transcript Normal curve - Beaufort County Schools

Normal Distribution
S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there
are data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
Measures of Central Tendency
 Mean
(𝝁): “average” add data values and
divide by the number of values
 Median:
list data values in order from least
to greatest and find the middle
 Mode:
most frequently occurring value
 Bimodal:
 Outlier:
when a set of data has two modes
a value that is substantially
different from the rest of the data in a set
Example
 Find
the mean, median and mode for the
sets of data. Identify any outliers.
 75,
68, 43, 120, 65, 180, 95, 225, 140
 3.4,
4.5, 2.3, 5.9, 9.8, 3.3, 2.1, 3.0, 2.9
Standard Deviation and Variance
 Measures
showing how much data values
deviate from the mean (spread)
 σ (sigma): standard deviation
 𝜎 2 (sigma squared): variance
 In
calculator:
 type
values into L1 (STAT – EDIT)
 STAT – CALC - #1 (gives all stats about that
data)
Normal Distributions
 Contains
data that varies randomly from
the mean
 EX:
test scores, weight, height, etc.
 Normal
curve: the graph of a normal
distribution
 Mean
in the middle
 “bell curve” shape
 3 standard deviations above and below the
mean
Skew
 Outliers
cause a normal curve to be right
skewed or left skewed
 If
model is skewed, measures of central tendency
are NOT the same
Empirical Rule “68-95-99.7”
 68%
of data between ± 1 standard deviations
of mean
 95% of data between ± 2 standard deviations
of mean
 99.7% of data between ± 3 standard deviations
of mean
Example
 The
number of hours that students studied for
final exams was normally distributed. Of the
200 students, the mean number of hours they
studied was 12 hours. The standard deviation
was 3 hours.
 Draw
a normal curve to represent the
distribution.
 What percentage of students studied 3 hours or
less?
 Of the students surveyed, how many studied
less than 9 hours?
Example
 The
heights of girls in a school choir are
distributed normally, with a mean of 64 and
a standard deviation of 1.75. If 83 girls are
between 60.5 in. and 67.5 in. tall, how
many girls are in the choir?
Z-score
 The
number of standard deviations a data
value is from the mean
 Formula:
 EX:
𝑧=
𝑥−𝜇
𝜎
A teacher gave a test that had a class
average of 85 with a standard deviation of 4.
If Sam scored a 90 on the test, how many
standard deviations from the mean is Sam’s
test score?
How to read a z-table

Refers to the standard normal curve (mean of 0
and standard deviation of 1)

Use the table to determine the area under the
curve or percentage of the area under the curve
 Numbers
on the top and sides represent z-scores
 Numbers
inside the table are areas/percentages
Types of areas using the z-table
 Areas
to the left: find z-value on table
EX: Given the z-score, find the area to the left
under the curve. z = 0.63
EX: Given the z-score, find the probability under
the normal curve for P(z < -1.45)
Types of areas using the z-table
 Areas
to the right: find z-value on table
and subtract from 1
EX: Find the area under the normal curve to the
right of z = -2.72
Types of areas using the z-table
 Areas
between two positive and two
negative: find two areas and subtract (larger –
smaller)
EX: Find the probability under the normal curve for
P (0.06 < z < 2.41)
Types of areas using the z-table
 Areas
between one positive and one
negative: subtract area to the left of the
negative from the area to the left of the
positive (larger – smaller)
EX: Find the area under the curve between
z = 0.77 and z = -0.82
Finding z-scores from data values
 Substitute
mean, standard deviation and
data value into z-score formula
 Simplify
to get z-score
EX: On a statistics test, the class mean was 63 with
a standard deviation of 7. Find the area under the
normal curve for a student who made a 73.
Percentiles

The percent of the population that is less than or
equal to a value (comparison to the rest of the data)
 Measures
position from the minimum
Percentiles

EX: A normal distribution of test scores has a
mean of 83 and a standard deviation of 6.
Everyone scoring at or above the 80th percentile
gets placed in an advanced class. What is the
cutoff score to get into the class?

EX: Suppose that you enter a fishing contest. The
contest takes place in a pond where the fish
lengths have a normal distribution with mean 16
inches and standard deviation 4 inches. Now
suppose you want to know what length marks the
bottom 10 percent of all the fish lengths in the
pond. What percentile are you looking for?