1. PRESENTATION OF DATA

2. Mathematical Presentation
Measures of Central Location

3. Learning Objectives Given a set of data, Identify the mode
Determine the median
Calculate the mean
Choose which measure of central location is most appropriate for the data

4. Types of Measures Central Location / Position / Tendency –
a single value that represents (is a good summary of) an entire distribution of data
Spread / Dispersion / Variability –
how much the distribution is spread or dispersed from that from its central location

5. Measures of Central Tendency
Parameter:
Descriptive measurement
computed from data of population
Statistic:
descriptive measurement
computed from data of a sample

6. Measure of Central Location
Definition: a single value that represents (is a good summary of) an entire distribution of data
Also known as:
“Measure of central tendency” “Measure of central position”
Common measures Arithmetic mean Median
Mode

7. MEAN Arithmetic mean: The sum of all
value of a set of observation divided by the number of these observations

8. MEAN Characteristics of the Mean: A single value
understand
It take in consideration all values in the set ( did not exclude any single value)
Greatly affected by extreme value(s)
Characteristics of the Mean:
A single value
Simple, easy to compute
and to

9. MEAN Calculated by this equation: ∑ x Mean of population μ =----------

10. Arithmetic Mean 605   30.25 20 N = 20 xi = 605 Obs Age 1 27 2 3 284 5 6 29 7 8 9 10 30 11 12 13 14 15 31 16 17 32 18 34 19 36 20 37
Arithmetic Mean
N = 20
xi = 605
605  
30.25 20

11. Mean Uses All Data, So Sensitive to Outliers6 5 4 3 2 1
Mean = 12.0 5 10 15
20 25
Nights of stay 30 35 40 45 50 6
Mean = 15.3 5
Numbe 4 3 2 1
Nights of stay

12. When to use the arithmetic mean?
Centered distribution Approximately symmetrical Few extreme values (outliers)
OK!

13. MEAN Weighted mean: the individual
values in the set are weighted by their respective frequencies.

14. Median
Definition: Median is the middle value; also, the value that splits the distribution into two equal parts
50% of observations are below the median
50% of observations are above the median
Method for identification Arrange observations in order Find middle position as (n + 1) / 2
1. Identify the value at the middle

15. MEDIAN
After creating ordered array (arranging data in an ascending or descending order), the median will be the middle value that divides the set of observations into two equal halves.

16. MEDIAN Characterized by: A single value
Simple, easy to compute , and easy to understand
Did not take in consideration all observations
Not affected by extreme values

17. MEDIAN Steps in computing the median: Create ordered array
Find position of the median which depends on the number of observation in the set:
If it is odd no.:
position of median= (n+1)/2, the median value is then specified

18. MEDIAN
If it is odd no.:
position of median= (n+1)/2, the median value is then specified

19. MEDIAN If it is even no. we will have 2 positions of the median:
n/2 & n/2+1, the median will be the mean of the two middle values

20. Median: Odd Number of Values
Obs
Age 1 27 2 3 28 4 5 6 29 7 8 9 10 30 11 12 13 14 15 31 16 17 32 18 34 19 36
Median:
Odd Number of Values
N = 19
N+1 2
19+1 20 10
Median Observation = = = =
Median age = 30 years

21. Median: Even Number of Values
Obs
Age 1 27 2 3 28 4 5 6 29 7 8 9 10 30 11 12 13 14 15 31 16 17 32 18 34 19 36 20 37
Median:
Even Number of Values
N = 20
N+1
Median Observation = 2
20+1 = 2 21 = 2
10.5 =
Median age = Average value between 10th and 11th observation
30+30 2
30 years =

22. Find Median of Length of Stay Data; Is Median Sensitive to Outliers?
0, 2, 3, 4, 5,
9, 9, 10, 10, 10,
12, 13, 14, 16, 18,
5, 6, 7, 8, 9,
10, 10, 11, 12, 12,
18, 19, 22, 27, 49
18, 19, 22, 27, 149

23. Median at 50%
= 10

24. MODE The most frequently occurring value in a series of observations.
Data distribution with one mode is called
unimodal; two modes is called bimodal;
more than distribution. nonmodal
two is called multimodal
Sometimes the data is

25. Mode Definition: Mode most frequently is the value that occurs
Method for identification
Arrange data into frequency distribution or histogram, showing the values of the variable and the frequency with which each value occurs
Identify the value that occurs most often

26. MODE
To determine the mode in a set of large number of observations, it may be mandatory to create a table showing the frequency distribution of observations values. The most frequent value will be the mode.

27. Mode Mode Age Frequency 27 2 28 3 29 4 30 5 31 32 1 33 34 35 36 37
Ob s
Age 1 27 2 3 28 4 5 6 29 7 8 9 10 30 11 12 13 14 15 31 16 17 32 18 34 19 36 20 37
Mode
Age
Frequency 27 2 28 3 29 4 30 5 31 32 1 33 34 35 36 37
Total 20
Mode

28. Mode Mode = 30 The most frequent value of the variable 7 6 5 4 3 2 1 2
Obs
Age 1 27 2 3 28 4 5 6 29 7 8 9 10 30 11 12 13 14 15 31 16 17 32 18 34 19 36 20 37
Mode
The most frequent value of the variable
Mode = 30 7 6 5 4 3 2 1
Frequency 2
Age (years)

29. Finding Mode from Length of Stay Data
0, 2, 3, 4, 5, 5, 6, 7, 8, 9,
9, 9, 10, 10, 10, 10, 10, 11, 12, 12,
12, 13, 14, 16, 18, 18, 19, 22, 27, 49

30. Mode = 10

31. Finding Mode from Histogram6 5 4 3 2 1
Number 5 10 15
Nights of stay 35 40 45 50

32. Mode – Sensitive to Outliers?6 5 4 3 2 1
Number o
Nights of stay

33. Unimodal Distribution20 18 16 14 12 10 8 6 4 2
Population
Bimodal Distribution 18 16 14 12 10 8 6 4 2
Population

34. Mode – Properties / Uses
Easiest measure to understand, explain, identify
Always equals an original value
Insensitive to extreme values (outliers) •
Good descriptive measure, properties
May be more than one mode
May be no mode
Does not use all the data
but poor statistical

35. Exercise 141 Pat. no 1 5 9 13 2 10 7 3 11 4 12 15 6 14 8 T A sample
Distance
(mile)(X) 1 5 9 13 2 10 7 3 11 4 12 15 6 14 8 T
141
A sample
of 15
patients
making to a health
traveled
visits center these miles,
distances in
calculate
measures of central tendency.

36. ANSWER
∑ x
• Mean= = 141/15= 1.4 mile n

37. ANSWER
Median:
Arrange data in order: 3,3,5,5,6,7,9,11,12,12,12,13,13,15,15
Find the site of the median
Since (n=15) is odd number, then the site of the median will be
= n+1/2=8
So the median is the 8th value in the ordered array =11 mile

38. ANSWER x f 3 2 5 6 1 7 9 11 12 13 15
Total
Mode:
Create a table of frequency distribution of observations in the set:
So the mode will be 12 mile since this value had the highest frequency

39. EXERCISE:
The mean age in months of preschool children in five villages are presented down; calculate the weighted mean of preschool children in these villages

40. EXERCISE: 1 44 58 2 78 45 3 48 62 4 60 5 47 59 village No. of children
Mean age (months) 1 44 58 2 78 45 3 48 62 4 60 5 47 59

41. ANSWER
Step 1: multiply the mean age for each village by the corresponding number of children in each village and add up the totals:
(44 X58)+ (78 X 45) + (48 X 62) + (45 X
60) + (47 X 59)
=14511 months

42. ANSWER Step 2: divide the total cumulative age by the
total number of children in the five villages
• = =
=55.38 months

43. Measures of Central Location – Summary
Measure of Central Location – single measure that represents an entire distribution
Mode – most common value
Median – central value
Arithmetic mean – average value
Mean uses all data, so sensitive to outliers
Mean has best statistical properties
Mean preferred for normally distributed data
Median preferred for skewed data