This document discusses key concepts in descriptive statistics including measures of dispersion like range and standard deviation, frequency distributions, the normal distribution, and the empirical rule. It defines range as the difference between the largest and smallest values. Frequency distributions provide the number of occurrences of values using tables or graphs. The normal distribution is widely used and defined by the mean and standard deviation. Standard deviation measures how far values deviate from the mean. The empirical rule states that 68%, 95%, and 99.7% of values in a normal distribution fall within 1, 2, and 3 standard deviations of the mean, respectively.

1. Descriptive
Statistics -2
Dr Qurat-Ul-Ain

2. Learning Objectives
MEASURE OF DISPERSION 1- RANG
2-FREQUENCY DISTRIBUTION
2- NORMAL DISTRIBUTION
STANDARD DEVIATION

3. RANGE
•Difference between the smallest and largest
observation/value
Example
10, 44, 45, 23, 86, 9
Range= 86-9=77

4. FREQUENCY DISTRIBUTION
•Frequency distribution in statistics
provides the information of the number
of occurrences (frequency) of distinct
values distributed within a given period
of time or interval, in a list, table, or
graphical representation.

5. FREQUENCY DISTRIBUTION
• example of the heights of ten students in cms.
• Frequency Distribution Table
139, 145, 150, 145, 136, 150, 152, 144, 138, 138

6. • Frequency Distribution Graph
• Using the same above example we can make the following
graph:

7. Types of Frequency Distribution
1. Grouped frequency distribution.
2. Ungrouped frequency distribution.
3. Cumulative frequency distribution.
4. Relative frequency distribution.
5. Relative cumulative frequency
distribution.

8. NORMAL DISTRIBUTION
•The normal distribution is the most
widely known and used of
all distributions. Because the normal
distribution approximates many
natural phenomena so well, it has
developed into a standard of
reference for many probability
problems.

9. NORMAL DISTRIBUTION
•Many groups follow this type of pattern.
That’s why it’s widely used in business,
statistics and in government bodies :
• Heights of people.
• Measurement errors.
• Blood pressure.
• Points on a test.
• IQ scores.
• Salaries.

10. NORMAL DISTRIBUTION
• The area under the normal curve is equal to
1.0.
• It is also known as the Gaussian distribution
and the bell curve.
• Denser in the center and less dense in the
tails.
• Defined by two parameters, the mean (μ) and
the standard deviation (σ).

11. STANDARD DEVIATION
• The standard deviation is the "average" degree
to which scores deviate from the mean. More
precisely, you measure how far all your
measurements are from the mean, square each
one, and add them all up. The result is called
the variance
• The square root of variance is standard
deviation

12. Formula of standard deviation

13. Height data are normally distributed. The
distribution in this example fits real data from
14-year-old girls during a study.

16. The Empirical Rule for the Normal
Distribution
• For example, in a normal distribution, 68% of the observations
fall within +/- 1 standard deviation from the mean. This
property is part of the Empirical Rule, which describes the
percentage of the data that fall within specific numbers of
standard deviations from the mean for bell-shaped curves.
• Mean +/- standard deviations ----- Percentage of data
contained
• 1 ------------ 68%
• 2 ------------ 95%
• 3 ------------ 99.7%

17. Mean +/- standard deviations -----Percentage of data contained
1 ------- 68%
2 ------- 95%
3 ------ 99.7%

18. For example
• Let’s look at a pizza delivery
example. Assume that a pizza
restaurant has a mean delivery time
of 30 minutes and a standard
deviation of 5 minutes. Using the
Empirical Rule, we can determine
that 68% of the delivery times are
between 25-35 minutes (30 +/- 5),
95% are between 20-40 minutes (30
+/- 2*5), and 99.7% are between 15-
45 minutes (30 +/-3*5).
• The chart illustrates this property
graphically.