PRESENTATION OF DATA

Mathematical Presentation Measures of Central Location

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

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

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

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

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

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

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

Arithmetic Mean 605   30.25 20 N = 20 xi = 605 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 Arithmetic Mean N = 20 xi = 605 605   30.25 20

Mean Uses All Data, So Sensitive to Outliers 6 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 0 10 20 30 40 50 60 70 80 90 100 110120 130140 150 Nights of stay

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

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

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

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.

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

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

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

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

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

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 =

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

Median at 50% = 10

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

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

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.

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

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 27 8 29 30 31 32 33 34 35 36 37 Age (years)

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

Mode = 10

Finding Mode from Histogram 6 5 4 3 2 1 Number 5 10 15 20 25 30 Nights of stay 35 40 45 50

Mode – Sensitive to Outliers? 6 5 4 3 2 1 Number o 0 10 20 30 40 50 60 70 80 90 100110120 130140150 Nights of stay

Unimodal Distribution 20 18 16 14 12 10 8 6 4 2 Population Bimodal Distribution 18 16 14 12 10 8 6 4 2 Population

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

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.

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

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

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

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

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

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

ANSWER Step 2: divide the total cumulative age by the total number of children in the five villages 14511 14511 • =--------------------------=------------- 44+78+48+45+47 262 =55.38 months

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