Descriptive stats

2. Outline
 Fundamentals of
Statistics
 Frequency distribution
 Group Data
 Ungroup Data
 Cumulative Frequency
 Graphical
Representation
 Histogram
 Polygon
 Ogive
 Pie Chart

3. Outline
 Measures of
Central
Tendency
 Mean
 Mode
 Median
 Measure of
Dispersion
 Range
 Variance
 Standard Deviation

4. Outline
 Measure of
Position
 Median
 Quartiles
 Deciles
 Percentiles
 Measure of
Distribution
 Skew-ness
 Krutosis
 z-score
 Errors

5. Definitions
A population is the collection of all outcomes, responses,
measurements, or counts that are of interest.
A sample is a subset, or part, of a population.
E X A M P L E
In a recent survey, 1500 adults in Pakistan were asked if
they thought there was solid evidence of global warming.
Eight hundred fifty-five of the adults said yes. Identify the
population and the sample.
Population = 1500
Sample data = 855 yes’s and 645 no’s.

6. Frequency Distributions
A table that organizes data values into classes or
intervals along with number of values that fall in
each class ( f ).
Ungrouped Frequency Distribution
for data sets with few different values. Each value
is in its own class.
Grouped Frequency Distribution
for data sets with many different values, which are
grouped together in the classes.

7. Grouped V/S Ungrouped
Frequency Distributions
Ungrouped Grouped
Courses Students Age of People
Taken f Voters
18-30
31-42
43-54
55-66
67-78
78-90
f
1
2
3
4
5
6
25 202
508
620
413
158
32
38
217
1462
932
15

8. Ungrouped Frequency Distributions
Number of 50 Players
play
different games
Players
f
Games
5
3
6
6
5
4
5
6
4
5
5
7
5
2
5
5
1
6
5
5
4
6
4
3
7
4
6
6
4
7
6
3
5
5
4
5
2
6
5
6
4
5
5
5
3
6
6
4
3
5
1
2
3
4
5
6
7
1
2
5
9
18
12
3

9. Grouped Frequency Distributions
Step 1. Find the minimum and maximum value
of data.
Step 2. Determine the range of the data.
Range = maximum value - minimum value
Step 3. Decide the number of classes/groups .
Number of classes should be between 5 and 20
Step 4. Find the class Interval.
Range
No.of
Classes
Class Interval  h 

10. Grouped Frequency Distributions
Step 5. Find the class limits. You can use the minimum
data entry as the lower limit of the first class. To find
the remaining lower limits, add the class width to the
lower limit of the preceding class. Then find the upper
limit of the first class. Remember that classes cannot
overlap. Find the remaining upper class limits.
Step 6. Make a tally mark for each data entry in the row
of the appropriate class.
Step 7. Count the tally marks to find the total frequency
f for each class.

11. Example
The following sample data set lists the prices of 30 portable
different sports equipment. Construct a frequency
distribution.
275 270 150 130 59 200 160 450 300 130 220 100 200 400 200 250 95 180
170 150 90 130 400 200 350 70 325 250 150 250
Class Tally Frequency
59–114 ||||
||||
||||
||||
5
115–170
171–226
227–282
283–338
339–394
395–450
|||
|
8
6
5
||
|
2
1
3
|||
∑ f = 30

12. Class Mark
The midpoint of a class is the sum of the lower and
upper limits of the class divided by two. The midpoint
is sometimes called the class mark.
Lower classlimit Upper class limit
X 
2

13. Solution
Class
Mark
Class Interval Frequency
59–114 5
8
6
5
2
1
3
86.5
115–170
171–226
227–282
283–338
339–394
395–450
142.5
198.5
254.5
310.5
366.5
422.5

14. Class boundaries
Class boundaries are the numbers that separate
classes without forming gaps between them. If data
entries are integers, subtract 0.5 from each lower limit
to find the lower class boundaries. To find the upper
class boundaries, add 0.5 to each upper limit. The
upper boundary of a class will equal the lower
boundary of the next higher class.
Classint erval
Class boundries  Mid point
2

15. Solution
Class
Interval
Class
Boundari
es
Frequency
59–11 4 5
8
6
5
2
1
3
58.5–114.5
114.5–170.5
170.5–226.5
226.5–282.5
282.5–338.5
338.5–394.5
394.5–450.5
115–170
171–226
227–282
283–338
339–394
395–450

16. Relative frequency
The relative frequency of a class is the portion or
percentage of the data that falls in that class. To
find the relative frequency of a class, divide the
frequency f by the sample size n.
class frequency f
relative frequency  
Sample size n

17. Solution
Class
Interval
Relative
frequency
Frequency
59–11 4 5
8
6
5
2
1
3
0.17
0.27
0.2
115–170
171–226
227–282
283–338
339–394
395–450
0.17
0.07
0.03
0.1

18. Graphical Representation
Pie Chart
A pie chart is a circle that is divided into sectors that
represent categories. The area of each sector is
proportional to the frequency of each category.
The following table shows the numbers of hours spent by
a Sport students on different events on a working day.
Measure of central
Activity No. of Hours
angle
Universit
y
Sleep
5
7
6
3
1
2
(5/24 × 360)° = 75°
(7/24 × 360)° = 105°
(6/24 × 360)° = 90°
(3/24 × 360)° = 45°
(1/24 × 360)° = 15°
(2/24 × 360)° = 30°
Playing
Study
T. V.
Others

19. Graphical Representation
Pie Chart

20. “Shape” of Distributions
Symmetric
Data is symmetric if the left half of its histogram is
roughly a mirror image of its right half.

21. “Shape” of Distributions
Skewed
Data is skewed if it is not symmetric and if it
extends more to one side than the other.

22. “Shape” of Distributions
Uniform
Data is uniform if it is equally distributed (on a
histogram, all the bars are the same height or
approximately the same height).

23. “Shape” of Distributions
Outliers
Unusual data values as compared to the rest of
the set. They may be distinguished by gaps in
a histogram.

24. Measures of Central Tendency
Measure of central tendency
A value that represents a typical, or central,
entry of a data set.
Most common measures of central tendency
 Mean
 Median
 Mode

25. Measure of Central Tendency:
Mean
The sum of all the data entries divided by the
number of entries.
x
 Sample mean=x 
n
Weights of 6 boys for weight lifting competition are 63, 57,
39, 41, 45, 45. Find the mean weight.
Number of observations = 6
Sum of all the observations = 63 + 57 + 39 + 41 + 45 + 45 = 290
Therefore, arithmetic mean = 290/6 = 48.3

26. Measure of Central Tendency:
Median
The median of a data set is the value that lies in the
middle of the data when the data set is ordered.
The median measures the center of an ordered data
set by dividing it into two equal parts. If the data set
has an
 odd number of entries: median is the middle
data entry.
 even number of entries: median is the mean of
the two middle data entries.

27. Computing the Median
If the data set has an:
• odd number of entries: median is the middle data
entry: 2 5 6 11 13
median is the exact middle value: x  6
• even number of entries: median is the mean of the
two middle data entries:
2 5 6 7 11 13
6  7
x   6.5
median is the mean of the by two numbers:
2

28. Measure of Central Tendency:
Mode
 The data entry that occurs with the greatest
frequency.
 If no entry is repeated the data set has no mode.
 If two entries occur with the same greatest
frequency, each entry is a mode (bimodal).
Mode is 1.10
a) 5.40 1.10 0.42 0.73 0.48 1.10
b) 27 27 27 55 55 55 88 88 99
c) 1 2 3 6 7 8 9 10
Bimodal - 27 & 55
No Mode

29. Mean v/s Median v/s Mode
All three measures describe an “average”. Choose the one
that best represents a “typical” value in the set.
 Mean:
 The most familiar average.
 A reliable measure because it takes into account every
entry of a data set.
 May be greatly affected by outliers or skew.
 Median:
 A common average.
 Not as effected by skew or outliers.
 Mode:
May be used if there is a vast repetition

30. Measures of Dispersion
Another important characteristic of quantitative data is
how much the data varies.
The most common methods for measuring of dispersion
are:
 Range
 Variance
 Standard deviation

31. Measures of Dispersion
Range
The difference between the maximum and
minimum data entries in the set.
 The data must be quantitative.
 Range = (Max. data entry) – (Min. data entry)
The scores of Pakistan cricket team in 1st test match
37 138 59 41 14 34 02 44 05 07. Find the range of
scores.
Range = (Max. scores) – (Min. scores)..
= 138-02=136

32. Measures of Position
In this section, you will learn how to use
fractiles to specify the position of a data entry
within a data set.
Fractiles are numbers that partition, or
divide, an ordered data set into equal parts.
For instance, the median is a fractile
because it divides an ordered data set into
two equal parts.

33. Measures of Position
the Quartiles is a fractile because it divides
an ordered data set into four equal parts.
the Deciles is a fractile because it divides an
ordered data set into ten equal parts.
the Percentiles is a fractile because it divides
an ordered data set into hundred equal parts.

34. Measures of Distribution
A fundamental task in many statistical analyses is
to characterize the location and variability of a
data set. A further characterization of the data
includes
skewness
kurtosis.

35. Measures of Distribution
Skewness tells us about the direction of variation
of the data set.
Kurtosis is a measure of whether the data are:
heavy-tailed or
light-tailed or
relative to a normal distribution.

36. Measures of Distribution
Skewness

37. Measures of Distribution
Skewness

38. References
1. Ron Larson
Elementary statistics: picturing the world
Pearson Education, 2012
2. David Miller
Measurement by the Physical Educator Why
and How
McGraw-Hill Higher Education, 2013

39. The Empirical Rule