Measures of Location and Dispersion Central Tendency

Measures of Location and

Dispersion

Measures of Location and

Dispersion

Central TendencyCentral TendencyCentral TendencyCentral Tendency

Mean, Median, ModeMean, Median, ModeMean, Median, ModeMean, Median, Mode

Of Statistical DataOf Statistical Data

•where µ is the population mean.

•X is a particular value. indicates the operation of adding.

•N is the number of observations in the population.

Population MeanPopulation MeanPopulation MeanPopulation Mean

• The population mean is the sum of the population values divided by the number of population values:

NX /

PRACTICE PRACTICE PRACTICE PRACTICE

• Parameter: a measurable characteristic of a population.

• The scores of 4 low-achieving students is compared: 56; 23; 42; and 73. Find the mean score.

• The mean is: (56+ 23+ 42+ 73)/4 = 48.5

NX /

Sample MeanSample MeanSample MeanSample Mean

•The sample mean is the sum of the sample values divided by the number of sample values:

X X n /

•where X is for the sample mean

• n is the number of observations in the sample

PRACTICPRACTICEE

PRACTICPRACTICEE

• A sample of five part-time teacher salaries is compared: $15,000, $15,000, $17,000, $16,000, and $14,000. Find the mean bonus for these five teachers.

• Since these values represent a sample size of 5, the sample mean is (14,000 + 15,000 +17,000 + 16,000 +15,000)/5 = $15,400.

X X n /

Properties of the Arithmetic Properties of the Arithmetic MeanMean


1. Every set of interval and ratio-level data has a mean

2. All the values are included in computing the mean

3. A data set will only have one unique mean

4. The mean is a useful measure for comparing two or more populations (we'll discuss this one later when we get to t-tests and analysis of variance)and…



5. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.

( )X X 0

PRACTICEPRACTICEPRACTICEPRACTICE

•What is the mean for: 3, 8, and 4?

•The mean is 5. •Illustrate the fifth

property:( )X X 0(3-5) + (8-5) + (4-5) = -2 +3 -1 = 0.

The MedianThe MedianThe MedianThe Median

• Median: The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest.

• There are as many values above the median as below it in the data array.

• Note: For an even set of numbers, the median will be the arithmetic mean of the two middle numbers.


• Compute the median for the following data:• The age of a sample of five college students

is: 21, 25, 19, 20, and 22.• Arranging the data in ascending order gives:

19, 20, 21, 22, 25. Thus the median is 21.• The height of four basketball players, in

inches, is 76, 73, 80, and 75.• Arranging the data in ascending order gives:

73, 75, 76, 80. Thus the median is 75.5.

Properties of the MedianProperties of the MedianProperties of the MedianProperties of the Median

1. There is a unique median for each data set.

2. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

3. It can be computed for ratio-level, interval-level, and ordinal-level data.

4. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

The ModeThe ModeThe ModeThe Mode

• The mode is the value of the observation that appears most frequently.

• EXAMPLE 6: the exam scores for ten students are: 83, 89, 84, 75, 99, 87, 83, 75, 83, 87. Find the mode.

• Since the score of 83 occurs the most, the modal score is 83.

Measures of DispersionMeasures of DispersionMeasures of DispersionMeasures of Dispersion

Calculating Data SPREADCalculating Data SPREAD

Measures of Measures of DispersionDispersion

Measures of Measures of DispersionDispersion

• The range is the difference between the highest and lowest values in a set of data.

• RANGE = Highest Value - Lowest Value

• PRACTICE: A sample of five accounting graduates revealed the following starting salaries: $22,000, $28,000, $31,000, $23,000, $24,000. The range is $31,000 - $22,000 = $9,000.

Population Variance Population Variance Population Variance Population Variance

•The population variance is the arithmetic mean of the squared deviations from the population mean.

2

2

( )X

N


• The ages of the Jones family are 2, 18, 34, and 42 years. What is the population variance?

X N/ /96 4 24

236

4

944

4

)2442(...242

/)(22

22

NX

3-35

2

2

( )X

N

Population VariancePopulation VariancePopulation VariancePopulation Variance

•Alternative formulas for the population variance are:

222

N

X

N

X

NN

XX

22

2

)(

The Population Standard The Population Standard DeviationDeviation

The Population Standard The Population Standard DeviationDeviation

• The population standard deviation (σ) is the square root of the population variance.

• For the PRACTICE, what is the standard deviation for the ages? (variance = 236)

• the population standard deviation is 15.36 (square root of 236).

1

)(

s

1

)(s

22

2

22

nnX

X

n

XX

Sample VarianceSample VarianceSample VarianceSample Variance

•The sample variance estimates the population variance. Conceptual Formula

Computational Formula

1

)(

s = Formula nalComputatio

22

2

n

n

XX


• (295-1369/5)/5-1• s2 = 21.2/(5-1) = 5.3

•A sample of five hourly wages for various jobs on campus is: $7, $5, $11, $8, $6. Find the variance.

Sample Standard Sample Standard DeviationDeviation

Sample Standard Sample Standard DeviationDeviation

•The sample standard deviation is the square root of the sample variance.

• Find the sample standard deviation for example 14. s2 = 5.3

•s = 2.30

Interpretation and Uses of Interpretation and Uses of the Standard Deviationthe Standard Deviation

Interpretation and Uses of Interpretation and Uses of the Standard Deviationthe Standard Deviation

• Empirical Rule: For any symmetrical, bell-shaped distribution, approximately 68% of the observations will lie within +/- one SD of the mean approximately 95% of the observations will lie within +/- 2 SD of the mean approximately 99.7% within +/- 3 SD of the mean.

. and between iprelationsh theshowing Curve Shaped-Bell

Documents

Measures of Location and Dispersion Central Tendency