This document discusses frequency distributions, histograms, and the normal distribution. It provides examples of grouped and relative frequency distributions and how to construct histograms to visualize this data. It also explains key properties of the normal distribution including the empirical rule and how it relates to standard deviations from the mean. Finally, it covers how to calculate z-scores to standardize values and use z-tables to find probabilities for the standard normal distribution.

1. NORMAL
DISTRIBUTIONS

2. FREQUENCY DISTRIBUTIONS AND
HISTOGRAMS

3. FREQUENCY DISTRIBUTIONS AND HISTOGRAMS
➢Large sets of data are often displayed using a frequency
distribution or a histogram.For instance,consider the following
situation.
STUDENT SCORE PERCENT OF STUDENTS
0-5 8.0
5-10 16.0
10-15 18.0
15-20 9.0
20-25 12.0
25-30 18.0
➢ CLASS- The number of observations that occur
in a particular predefined interval.
➢ UPPER CLASS BOUNDARY-The highest end
point of an open interval
➢ LOWER CLASS BOUNDARY-The lowest starting
point of an open interval.
➢ INTERVAL-Interval data is measured along a
numerical scale that has equal distances
between adjacent values.

4. ➢ A histogram is a graph that shows the frequency of numerical data using rectangles. The
height of a rectangle (the vertical axis) represents the distribution frequency of a variable (the
amount, or how often that variable appears).
HISTOGRAM
0
2
4
6
8
10
12
14
16
18
20
0-5 5 10 10 15 15-20 20-25 25-30
PERCENT
OF
STUDENTS
STUDENT SCORE

5. TWO TYPES OF FREQUENCY DISTRIBUTIONS
1.GROUPED FREQUENCY DISTRIBUTION
➢Grouped frequency is the frequency where
several numbers are grouped together. Grouped
frequency distribution helps to organize the data
more clearly. It is more useful when the scores
have multiple values.

6. GROUPED FREQUENCY DISTRIBUTION
EXAMPLE:
The data represent the ages of 28 men when they get a gift construct a group
frequency with 7 classes and create a histogram.
3 29 15 16 22 8 9 13 16 27 3 9 26 23 12 14 19 26 30 7 6 19 20 30 27 19 20
0
1
2
3
4
5
6
3 6 7 10 11 14 15 18 19 22 23 26 27 30
NUMBER
OF
MEN
AGES OF MEN
AGES OF MEN
(CLASS LIMIT)
NUMBER OF
MEN(FREQUENCY)
3-6 5
7-10 4
11-14 3
15-18 3
19-22 5
23-26 3
27-30 5

7. 2. RELATIVE FREQUENCY DISTRIBUTION
➢A relative frequency distribution shows the
proportion of the total number of observations
associated with each value or class of values and
is related to a probability distribution, which is
extensively used in statistics.
TWO TYPES OF FREQUENCY DISTRIBUTIONS

8. RELATIVE FREQUENCY DISTRIBUTION
Example:The data represent the average score of 45 student,construct a relative
frequency with 7 classes and histogram.
2 12 5 10 8 9 11 4 18 3 13 6 11 9 8 14 5 20 4 14 7 3 10 7 16 6 22 15 2 3 6 16 8 17 9 18
19 4 5 20 7 10 11 12 13
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 5 7 8 10 11 13 14 16 17 19 20 22
PERCENT
OF
SCORES
NUMBER OF STUDENT
NUMBER OF
STUDENT(C
LASS LIMIT
)
PERCENT
OF
SCORES(FR
EQUENCY)
RELATIVE
FREQUENC
Y
2-4 8 0.8
5-7 9 0.9
8-10 9 0.9
11-13 7 0.7
14-16 5 0.5
17-19 4 0.4
20-22 3 0.3
TOTAL: 45 TOTAL: 4.5

9. NORMAL DISTRIBUTION AND
THE EMPIRICAL RULE

10. NORMAL DISTRIBUTION AND THE EMPIRICAL RULE
➢One of the most important statistical distribution of data
is known as a Normal Distribution . This distribution
occurs in a variety of a application.
➢ A Normal Distribution a continuous probability
distribution of data that has the shape of symmetrical-
bell curve.
➢ It is also called as GAUSSIAN DISTRIBUTION/CURVE,
named after Carl Gauss.

11. ➢PARAMETERS OF NORMAL CURVE
1) 𝜋 – Mean
2) 𝜎 – Standard deviation

12. PROPERTIES OF STANDARD DEVIATION
I. Bell-shaped and symmetrical with respect to the
vertical axis.
II.The mean, median, and mode is equal.
III.Asymptotic with respect to the horizontal axis.
IV.Total area under the curve is 1. The area represents
the percentage/probability under the curve.
V.The width of the curve is determined by the standard
deviation of the normal distribution

13. EMPIRICAL RULE
➢ Also known as the Three Sigma Rule
➢ It is statistical rule which states that for a normal
distribution, almost all observed data will fall within 3
standard deviation from the mean.
❖68% of all data set falls within 1 SD (𝜋 ± 𝜎) from the
mean.
❖95% of all data set falls within 2 SD (𝜋 ± 2𝜎) from the
mean.
❖99.7% of all data set falls within 3 SD (𝜋 ± 3𝜎) from the
mean.
▪ Lacking: 100 – 99.7 = .3% – outliers

14. EMPERICAL RULE IN BELL CURVE

15. Example 1. The average IQ score is 100 with Standard
Deviation of 15.
1) What score fall within 68% of the distribution?
2) What score fall within 95% of the distribution?
3) What score fall within 99.7% of the distribution?
USE THE IMPERIAL RULE TO SOLVE AN APPLICATION

16. Example 2. The average score of SHS students in your school is
normally distributed with a mean of 84 and a standard deviation of 3.
1) What average grades fall within 68% of the distribution?
2) If your average grade is 90, how many SD are you away from the
mean?
3) What percent of the average are between 78 to 87?
USE THE IMPERIAL RULE TO SOLVE AN APPLICATION

17. Example 3. A survey of 1,000 U.S. gas stations found that the
price charged for a gallon of regular gas could be closely
approximated a normal distribution with a mean of $3.10 and a
standard deviation of $0.18. How many of the station charge?
1) Between $2.74 and $3.46 for a gallon of regular gas?
2) Less than $3.28 for a gallon of regular gas?
3) More than $3.46 for a gallon of regular gas?
USE THE IMPERIAL RULE TO SOLVE AN APPLICATION

18. THE STANDARD NORMAL
DISTRIBUTION

19. ➢also known as the “z-distribution.
➢It is the normal distribution that has a mean of 0 and a standard deviation of
1.It visualizes the distribution of a set of chosen values across a specified
group that tend to have central,normal values,as peak with low and high
extremes tapering off relatively symmetrically on either side.
THE STANDARD NORMAL DISTRIBUTION

20. ➢ These are unimodal and
symmetrically distributed with a
bell-shaped curve.
➢ It can take on any value as it means
and standard deviation.
➢ It is where the mean and the
standard deviation is always fixed.
When (x) lies:
➢ A positive z-score means that your
x-value is greater than the mean.
➢ A negative z-score means that your
x-value is less than the mean.
➢ A z-score of zero means that your x-
value is equal to the mean.
NORMAL DISTRIBUTION

21. *To standardize a value from a normal
distribution,convert the individual value into a z-score.
1. Subtract a mean from your individual value.
2. Divide the difference by the standard deviation.
Where:
x= individual value
ó=standard deviation
HOW TO CALCULATE A Z-SCORE?

22. ➢ You collected SAT score form student in preparation in a new test
preparation course .The data follows a normal distribution with a mean
score of 1150 and a standard deviation of 150.You want to find the
probability that SAT scores exceed 1380.
Step 1 . Subtract mean from individual value
X=1380
M-1150
1380-1150 = 230
Step 2. Divide the difference by the standard deviation
Standard deviation =150
Z= 230/150
Z= 1 .53
EXAMPLE

23. ➢ z-table tells you the total are under the curve upto a given
z-score.
➢ The first column contains z-score up to first decimal places
➢ The top row of the table gives the second decimal place.
To find the corresponding area under the curve
(Probability)for a z-score.(Table)
➢ Go down to the row with the first two digits of your z-
score.
➢ Go across to the column with the same third digit as your
z-score
➢ Find the value at the intersection of the row and column
HOW TO USE Z-TABLE

24. Z-TABLE