1. B.A. –III Paper-III Unit-III Tabulation and Interpretation of DATA Measures of Central Tendency Measures of Dispersion
By-
Dr. Ankita Gupta
Asstt. Professor
Economics Department
Mahatma Gandhi Kashi Vidyapith

2. Summarizing data Tables Charts and graphs
Simplest way to summarize data
Data are presented as absolute numbers or percentages
Charts and graphs
Visual representation of data
The two main ways of summarizing data are by using tables and charts or graphs.
A table is the simplest way of summarizing a set of observations. A table has rows and columns containing data, which can be in the form of absolute numbers or percentages, or both.
Charts and graphs are visual representations of numerical data and, if well designed, will convey the general patterns of the data.

3. Basic guidance when summarizing data
Ensure graphic has a title
Label the components of your graphic
Indicate source of data with date
Provide number of observations (n=xx) as a reference point
Add footnote if more information is needed
To make your graphics as self-explanatory as possible, there are several things to always include:
Every table or graph should have a title or heading
The x- and y-axes of a graph should be labeled – include value labels, such as a percentage sign; include a legend
Always cite the source of your data and put the date of data were collection or publication
Provide the sample size or the number of people to which the graph is referring (N)
Include a footnote if the graphic isn’t self-explanatory
These points will pre-empt questions and explain the data. In the next several slides, we’ll see examples of these points.

4. Basic guidance when summarizing data
Ensure graphic has a title
Label the components of your graphic
Indicate source of data with date
Provide number of observations (n=xx) as a reference point
Add footnote if more information is needed
To make your graphics as self-explanatory as possible, there are several things to always include:
Every table or graph should have a title or heading
The x- and y-axes of a graph should be labeled – include value labels, such as a percentage sign; include a legend
Always cite the source of your data and put the date of data were collection or publication
Provide the sample size or the number of people to which the graph is referring (N)
Include a footnote if the graphic isn’t self-explanatory
These points will pre-empt questions and explain the data. In the next several slides, we’ll see examples of these points.

5. Charts and graphs Charts and graphs are used to portray:
Trends, relationships, and comparisons
The most informative are simple and self- explanatory
Although they are easier to read than tables, charts provide less detail. The loss of detail may be replaced by a better understanding of the data.

6. Use the right type of graphic
Charts and graphs
Bar chart: comparisons, categories of data
Line graph: display trends over time
Pie chart: show percentages or proportional share
We’re going to review the most commonly used charts and graphs in Excel/PowerPoint. Later, we’ll have you use data to create your own graphics, which may go beyond those presented here.
Bar charts are used to compare data across categories.
Line graphs are used to display trends over time.
Pie charts show percentages or the contribution of each value to a total.

7. Bar chart Comparing categories
In this bar chart, we’re comparing the categories of data, which are the different sites. You see a comparison between sites by quarters and between quarters over time.
What should be added to this chart to provide the reader with more information?
NOTE to facilitator: Wait for a participant response before answering (and then show next slide).
On the next slide, we see how the graph has been improved and is now self-explanatory.

8. Measures of Central Tendency
According to Prof Bowley “Measures of central tendency (averages) are statistical constants which enable us to comprehend in a single effort the significance of the whole.”
The main objectives of Measure of Central Tendency are –
1) To condense data in a single value.
2) To facilitate comparisons between data.

9. Requisites of a Good Measure of Central Tendency:
1. It should be rigidly defined.
2. It should be simple to understand & easy to calculate.
3. It should be based upon all values of given data.
4. It should be capable of further mathematical treatment.
5. It should have sampling stability.
6. It should be not be unduly affected by extreme values.

10. Conduct further research
Interpreting data
Adding meaning to information by making connections and comparisons and exploring causes and consequences
Relevance of finding
Reasons for finding
Consider other data
Conduct further research
Data interpretation is the process of making sense of the information. It allows us to ask: What does this information tell me about the program?
Here, you see a flow chart of the steps involved in interpreting data …
NOTE to facilitator: Read the steps outlined in the diagram.

11. Interpretation – relevance of finding
Does the indicator meet the target?
How far from the target is it?
How does it compare (to other time periods, other facilities)?
Are there any extreme highs and lows in the data?
When interpreting data and seeking the relevance of our findings, we may ask these questions:
NOTE to facilitator: Read slide.
Asking these questions will help you to put the data in the context of your program.

12. Interpretation – relevance of finding
Adding meaning to information by making connections and comparisons and exploring causes and consequences
Relevance of finding
Reasons for finding
Consider other data
Conduct further research
We start by wanting to know the relevance of our findings. Seeking the relevance of a finding is to:
NOTE to facilitator: Read slide.

13. Interpretation – possible causes?
Supplement with expert opinion
Others with knowledge of the program or target population
Relevance of finding
Reasons for finding
Consider other data
Conduct further research
When seeking potential reasons for the finding, we often will need additional information that will put our findings into the context of the program.
Supplementing the findings with expert opinion is a good way to do this. For example, talk to others with knowledge of the program or target population, who have in-depth knowledge about the subject matter, and get their opinions about possible causes.
For example, if your data show that you have not met your targets, you may want to know if:
the community is aware of the service? To answer this, you could talk to community leaders or other providers to get their opinions.
Sometimes ad hoc conversations with experts are insufficient. To get a more accurate explanation of your findings, you often will have to consider other data resources.

14. Interpretation – consider other data
Use routine service data to clarify questions
Calculate nurse-to-client ratio, review commodities data against client load, etc.
Use other data sources
Relevance of finding
Reasons for finding
Consider other data
Conduct further research
Let’s go back to the finding of ‘the program has not met its annual target’. Can we understand why this is happening by looking at other program indicators?
You may want to calculate the nurse-to-client ratio to determine if the facility is sufficiently staffed to meet the client load.
You also may want to review commodity data with client load to determine if there are shortages of commodities.
While it is important to consider other indicators in your analysis, remember – descriptive statistics do not show causality. In these cases, look at other data sources.

15. Measures of Central Tendency
Mode
Median
Mean (Average)

16. Measures of Central Tendency
Mode
Value of the distribution that occurs most frequently (i.e., largest category)
Only measure that can be used with nominal- level variables
Limitations:
Some distributions don’t have a mode
Most common score doesn’t necessarily mean “typical”
Often better off using proportions or percentages

17. Measures of Central Tendency

18. Measures of Central Tendency
2. Median
value of the variable in the “middle” of the distribution
same as the 50th percentile
When N is odd #, median is middle case:
N=5:
median=6
When N is even #, median is the score between the middle 2 cases:
N=6:
median=(5+9)/2 = 7

19. MEDIAN: EQUAL NUMBER OF CASES ON EACH SIDE

20. Measures of Central Tendency
3. Mean
The arithmetic average
Amount each individual would get if the total were divided among all the individuals in a distribution
Symbolized as X
Formula: X = (Xi ) N

21. Measures of Central Tendency
Characteristics of the Mean:
It is the point around which all of the scores (Xi) cancel out. Example: X
(Xi – X) 3
3 – 7 -4 6
6 – 7 -1 9
9 – 7 2 11
11- 7 4
X = 35
(Xi – X) =

22. Measures of Central Tendency
Characteristics of the Mean:
2. Every score in a distribution affects the value of the mean
As a result, the mean is always pulled in the direction of extreme scores
Example of why it’s better to use MEDIAN family income
POSITIVELY SKEWED
NEGATIVELY SKEWED

23. Measures of Central Tendency
In-class exercise:
Find the mode, median & mean of the following numbers: 2
Does this distribution have a positive or negative skew?
Answers:
Mode (most common) = 2
Median (middle value) ( )= 4.5
Mean = (Xi ) / N = 51/10 = 5.1

24. Measures of Central Tendency
Levels of Measurement
Nominal
Mode only (categories defy ranking)
Often, percent or proportion better
Ordinal
Mode or Median (typically, median preferred)
Interval/Ratio
Mode, Median, or Mean
Mean if skew/outlier not a big problem (judgment call)

25. Measures of Dispersion
provide information about the amount of variety or heterogeneity within a distribution of scores
Necessary to include them w/measures of central tendency when describing a distribution

26. Measures of Dispersion
Range (R)
The scale distance between the highest and lowest score
R = (high score-low score)
Simplest and most straightforward measure of dispersion
Limitation: even one extreme score can throw off our understanding of dispersion

27. Measures of Dispersion
2. Interquartile Range (Q)
The distance from the third quartile to the first quartile (the middle 50% of cases in a distribution)
Q = Q3 – Q1
Q3 = 75% quartile
Q1 = 25% quartile
795

28. MEASURES OF DISPERSION
Problem with both R & Q:
Calculation based on only 2 scores
Not a very adequate representative measure

29. MEASURES OF DISPERSION
Standard deviation
Uses every score in the distribution
Measures the standard or typical distance from the mean
Deviation score = Xi - X
Example: with Mean= 50 and Xi = 53, the deviation score is = 3

30. The Problem with Summing Devaitions From Mean
2 parts to a deviation score: the sign and the number
Deviation scores
add up to zero
Because sum of deviations
is always 0, it can’t be
used as a measure of
dispersion
X Xi - X
8 +5
1 -2
3 0
0 -3
12 0
Mean = 3 

31. Average Deviation (using absolute value of deviations)
Works OK, but…
AD =  |Xi – X| N
X |Xi – X|
AD = 10 / 4 = 2.5
X = 3
Absolute Value to get rid of negative values (otherwise it would add to zero)

32. Variance & Standard Deviation
Purpose: Both indicate “spread” of scores in a distribution
Calculated using deviation scores
Difference between the mean & each individual score in distribution
To avoid getting a sum of zero, deviation scores are squared before they are added up.
Variance (s2)=sum of squared deviations / N
Standard deviation
Square root of the variance Xi
(Xi – X)
(Xi - X)2 5 1 2 -2 4 6
 = 20
 = 0
 = 14

33. Terminology
“Sum of Squares” = Sum of Squared Deviations from the Mean =  (Xi - X)2
Variance = sum of squares divided by sample size =  (Xi - X)2 = s2 N
Standard Deviation = the square root of the variance = s

34. Calculation Exercise
Number of classes a sample of 5 students is taking:
Calculate the mean, variance & standard deviation
mean = 20 / 5 = 4
s2 (variance)= 14/5 = 2.8
s= 2.8 =1.67 Xi
(Xi – X)
(Xi - X)2 5 1 2 -2 4 6
 = 20 14

35. Calculating Variance, Then Standard Deviation
Number of credits a sample of 8 students is are taking:
Calculate the mean, variance & standard deviation Xi
(Xi – X)
(Xi - X)2 10 -4 16 9 -5 25 13 -1 1 17 3 15 2 4 14 18
 = 112 72

36. Summary Points about the Standard Deviation
Uses all the scores in the distribution
Provides a measure of the typical, or standard, distance from the mean
Increases in value as the distribution becomes more heterogeneous
Useful for making comparisons of variation between distributions
Becomes very important when we discuss the normal curve (Chapter 5, next)

37. Mean & Standard Deviation Together
Tell us a lot about the typical score & how the scores spread around that score
Useful for comparisons of distributions:
Example:
Class A: mean GPA 2.8, s = 0.3
Class B: mean GPA 3.3, s = 0.6
Mean & Standard Deviation Applet