Discussion on some descriptive tools identifying locations of data in a given data set

1. MEASURES OF POSITION
Descriptive Statistics

2. OBJECTIVES
Recognize, describe, calculate,
and interpret the measures of
location of data.

3. Measures of Location
•also known as measures of relative standing or
measures of spread, are statistical tools used to
determine where a particular data point falls within a
dataset.
•These measures help assess the position or
ranking of a value within the data
distribution

4. COMMON MEASURES OF LOCATION
•Quartiles, Deciles and
Percentiles •Kurtosis
•Skewness

5. COMMON MEASURES OF LOCATION
•Quartiles, Percentiles,
Percentile Rank

6. • common measures of location
are quartiles and percentiles
QUARTILE
• Quartiles are numbers that separate the data into
quarters (four parts). Like the median, quartiles may or
may not be an actual value in the set of data
• Quartiles divide the data into three points: Q1 (25th
percentile), Q2 (50th percentile, or median), and Q3
(75th percentile).

7. Calculation
Q1: The value at which 25% of the data falls below and
75% above.
Q2: The median, where 50% of the data is below and 50%
above.
Q3: The value at which 75% of the data falls below and
25% above.
QUARTILE

8. Median (Q2): The median, also known as the second quartile,
is the middle value in the dataset. It is a measure of central
tendency that is less sensitive to extreme outliers than the
mean. Q2 divides the data into two equal halves.
Explanation and Interpretation:
First Quartile (Q1): Q1, the 25th percentile, marks the point
below which 25% of the data falls. It represents the lower end
of the lower half of the dataset, providing information about
the spread of the lower values.

9. Third Quartile (Q3): Q3, the 75th percentile, marks the point
below which 75% of the data falls. It represents the upper end
of the upper half of the dataset, providing information about
the spread of the higher values.
Explanation and Interpretation:

10. Business Applications
Sales Performance: Quartiles can be used to assess the performance
of sales representatives by dividing them into quartiles based on
their sales figures. This helps identify top performers (Q4), average
performers (Q2 and Q3), and those who may need improvement (Q1).
Employee Performance: Quartiles can be used to assess and rank
employee performance based on various metrics such as sales
figures, customer satisfaction scores, or project completion times.
Employees in the top quartile may be recognized or rewarded, while
those in the bottom quartile may receive additional training or
coaching.

11. Business Applications
Product Performance: Quartiles can help analyze the performance
of different products or product categories. For instance, by
categorizing products into quartiles based on sales revenue,
businesses can identify their top-selling products and focus marketing
efforts or inventory management accordingly.
Customer Satisfaction: Quartiles can be used to analyze
customer survey data to identify areas of improvement. For
example, if customer satisfaction scores are divided into
quartiles, businesses can focus on addressing issues that are
most prevalent in the lowest quartile.

12. Considerations in Computing Quartiles
Quartile.Exc calculates quartiles using exclusive boundaries. This
means that when calculating Q1.Exc, it doesn't include the median
value in the lower 25% of the data. Similarly, for Q3.Exc, it doesn't
include the median value in the upper 25% of the data.
Quartile.Inc calculates quartiles using inclusive boundaries. This means
that when calculating Q1.Inc, it includes the median value in the lower
25% of the data. Similarly, for Q3.Inc, it includes the median value in
the upper 25% of the data.

13. 13
Add
a
foot
er

14. PERCENTILE

15. • Percentiles are statistical measures that divide a
dataset into 100 equal parts, with each part
representing a percentage of the data.
PERCENTILE
• They help you understand how a specific data point
compares to the entire dataset in terms of its relative
position
• Percentiles are often expressed as a percentage
and are commonly used to summarize and
interpret data.

16. • Sort the Data: Arrange your dataset in ascending
or descending order.
Calculation of Percentiles:
• Determine the Rank: Identify the rank (position) of the
desired percentile. For example, if you want to find the
25th percentile (Q1), this represents the value below
which 25% of the data falls.

17. Calculation of Percentiles:
• Interpolate If Necessary: If the rank is not an integer
(e.g., you're looking for the 75th percentile, which falls
between the 75th and 76th data points), you may need
to interpolate to estimate the exact value.
• Calculate the Percentile: Once you have the rank or
interpolated rank, you can find the corresponding value
in the sorted dataset. This value is the desired
percentile.

18. • Median (50th Percentile): The median is the middle value in a
dataset. It divides the data into two equal halves, with 50% of
the data below and 50% above it. The median is a measure of
central tendency and is less affected by extreme outliers
compared to the mean.
Common Percentiles and
Interpretation:
• Quartiles (25th, 50th, and 75th Percentiles): Quartiles divide the data
into four equal parts. The first quartile (Q1) is the 25th percentile, the
second quartile (Q2, which is also the median) is the 50th percentile, and
the third quartile (Q3) is the 75th percentile. They are used to analyze the
spread of data and identify outliers.

19. • Percentile Ranks (e.g., 90th Percentile): Percentile
ranks help you understand how a specific data point
compares to the rest of the dataset. For example, if a
student scores in the 90th percentile on a standardized
test, it means they scored higher than 90% of the test-
takers.
Common Percentiles and
Interpretation:

20. Application in Business Statistics:
Performance Benchmarking: Businesses can use percentiles
to set performance benchmarks. For instance, a company
might aim to have customer service response times in the top
10th percentile to ensure exceptional service quality.
Risk Assessment: In finance, percentiles are used to assess the
risk associated with investments. For example, the 5th percentile
of returns on an investment portfolio represents the lowest 5% of
possible returns, helping investors understand potential losses.

21. Application in Business Statistics:
Market Research: Percentiles are used to segment markets
based on income, spending habits, or other factors.
Companies can target specific percentile groups with tailored
marketing strategies.
Customer Segmentation: Percentiles can help businesses
identify and categorize their most valuable customers based
on factors like purchase frequency, transaction value, or
loyalty.

22. Application in Business Statistics:
Supply Chain Optimization: Companies can use percentiles to
analyse the variability in supply chain lead times and manage
inventory levels more effectively.
Quality Control: In manufacturing, percentiles are used to
ensure product quality. Products falling below a certain
percentile may be considered defective and subject to
further inspection or rejection.

23. Considerations in Computing Percentile
Percentile.Exc calculates percentiles using exclusive
boundaries. It does not include the specified
percentile value itself in the calculated percentage
of data below that percentile.
Percentile.Inc: Percentile.Inc calculates percentiles
using inclusive boundaries. It includes the specified
percentile value itself in the calculated percentage of
data below that percentile.

24. 24
Add
a
foot
er

25. SKEWNESS

26. Skewness
an important concept in statistics that helps us understand the
shape and symmetry of a data distribution
measure of the asymmetry of a probability distribution.
tells us whether the data is concentrated more on one side of the
mean than the other
Skewness is a crucial concept in descriptive statistics, especially when
analyzing datasets in fields like business, finance, economics, and social
sciences.

28. TYPES OF SKEWNESS
Positive Skewness (Right
Skewed): In a positively
skewed distribution, the tail
on the right-hand side
(greater values) is longer or
fatter than the left tail. This
means that the data has
outliers on the right side and
is concentrated towards the
left.

29. TYPES OF SKEWNESS
Negative Skewness (Left
Skewed): In a negatively
skewed distribution, the tail
on the left-hand side (smaller
values) is longer or fatter
than the right tail. This
indicates that the data has
outliers on the left side and is
concentrated towards the
right.

30. COMPUTATION OF SKEWNESS
The most common measure of skewness is the skewness
coefficient, denoted as "γ" or "Skew."
There are different formulas to compute skewness, but one of the commonly
used formulas is Pearson's First Coefficient of Skewness (Skewness coefficient):

31. COMPUTATION OF SKEWNESS
The skewness coefficient can be positive, negative, or zero,
indicating the direction and degree of skewness.
Positive skewness indicates right skew (tail to the right).
Negative skewness indicates left skew (tail to the left).
Zero skewness indicates a perfectly symmetrical distribution.

32. Interpreting Skewness:
The skewness coefficient can be positive, negative, or zero,
indicating the direction and degree of skewness.
Greater than +1 or less than -1 suggests a highly skewed distribution.
Between -1 and +1 suggests a moderately skewed distribution.
Close to 0 suggests a nearly symmetrical distribution.

33. 33
Add
a
foot
er