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Statistics Alan D. Smith Descriptive Statistics - Measures of Central Tendency

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- TO CALCULATE THE ARITHMETIC MEAN, THE WEIGHTED MEAN, THE MEDIAN, THE MODE, AND THE GEOMETRIC MEAN.
- TO EXPLAIN THE CHARACTERISTICS, USE, ADVANTAGES, AND DISADVANTAGES OF EACH MEASURE OF CENTRAL TENDENCY.
- TO IDENTIFY THE POSITION OF THE ARITHMETIC MEAN,MEDIAN, AND MODE FOR BOTH A SYMMETRICAL AND A SKEWED DISTRIBUTION.

- Definition:For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values. To compute the population mean, use the following formula.

Sigma

Individual

value

mu

Population

Size

EXAMPLE

- Parameter:A measurable characteristic of a population. For example, the population mean.
- A racing team has a fleet of four cars. The following are the miles covered by each car over their lives: 23,000, 17,000, 9,000, and 13,000. Find the average miles covered by each car.
- Since this fleet is the population, the mean is (23,000 + 17,000 + 9,000 + 13,000)/4 = 15,500.

- Definition:For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values. To compute the sample mean, use the following formula.

Sigma

X-bar

Individual

value

Sample

Size

EXAMPLE

- Statistic:A measurable characteristic of a sample. For example, the sample mean.
- A sampleof five executives received the following amounts of bonus last year: 14, 15, 17, 16, and 15 in $1,000. Find the average bonus for these five executives.
- Since these values represent a sample of size 5, the sample mean is (14,000 + 15,000 + 17,000 + 16,000 + 15,000)/5 = $15,400.

PROPERTIES OF THE ARITHMETIC MEAN

- Every set of interval-level and ratio-level data has a mean.
- All the values are included in computing the mean.
- A set of data has a unique mean.
- The mean is affected by unusually large or small data values.
- The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero.

Sum of Deviations

- Consider the set of values: 3, 8, and 4. The mean is 5. So (3 -5) + (8 - 5) + (4 - 5) = -2 + 3 - 1 = 0.
- Symbolically we write:
- The mean is also known as the “expected value” or “average.”

- Definition: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 odd set of numbers, the median will be the middle number in the ordered array.
- Note:For an even set of numbers, the median will be the arithmetic average of the two middle numbers.

- Compute the median:
- The road life for a sample of five tires in miles is: 42,000 51,000 40,000 39,000 48,000
- Arranging the data in ascending order gives: 39,000 40,000 42,000 48,000 51,000. Thus the median is 42,000 miles.

- Compute the median:
- The following values are years of service for a sample of six store managers: 16 12 8 15 7 23.
- Arranging in order gives 7 8 12 15 16 23. Thus the median is (12 + 15)/2 = 13.5 years.

- There is a unique median for each data set.
- It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.
- It can be computed for ratio-level, interval-level, and ordinal-level data.
- It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

THE MODE

- Definition:The mode is that value of the observation that appears most frequently
- The exam scores for ten students are: 81 93 75 68 87 81 75 81 87. What is the modal exam score?
- Since the score of 81 occurs the most, then the modal score is 81.
- The next slide shows the histogram with six classes for the water consumption from our previous class. Observe that the modal class is the blue box with a midpoint of 15.

THE WEIGHTED MEAN

- Definition:The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ... , wn, is computed from the following formula.
- Why would anyone want to weight observations?

EXAMPLE

- During a one hour period on a busy Friday night, fifty soft drinks were sold at the Kruzin Cafe. Compute the weighted mean of the price of the soft drinks. (Price ($), Number sold): (0.5, 5), (0.75, 15), (0.9, 15), and (1.10, 15).
- The weighted mean is

[0.5´5 + 0.75´15 + 0.9´15 + 1.1´15]/[5 +15+15+ 15] = $43.75/50 =

$0.875

THE GEOMETRIC MEAN

- Definition:The geometric mean (GM) of a set of n numbers is defined as the nthroot of the product of the n numbers. The formula for the geometric mean is given by:
- One main use of the geometric mean is to average percents.
- How do you compute the nth root of a number?

EXAMPLE

- The profits earned by ABC Construction on three projects were 6, 3, and 2 percent respectively. Compute the geometric mean profit and the arithmetic mean and compare.
- The geometric mean is
- The arithmetic mean profit =(6 + 3 + 2)/3 = 3.6667.
- The geometric mean of 3.3019 gives a more conservative profit figure than the arithmetic mean of 3.6667. This is because the GM is not heavily weighted by the profit of 6 percent.

THE GEOMETRIC MEAN (continued)

- The other main use of the geometric mean to determine the average percent increase in sales, production or other business or economic series from one time period to another.
- The formula for the geometric mean as applied to this type of problem is:
- Where did this come from?

EXAMPLE

- The total enrollment at a large university increased from 18,246 in 1985 to 22,840 in 1995. Compute the geometric mean rate of increase over the period.
- Here n = 10, so n - 1 = 9 = (number of periods)
- The geometric mean rate of increase is given by
- That is, the geometric mean rate of increase is 2.53%.

EXAMPLE

- Do you prefer I use the arithmetic mean or the geometric mean to compute class score averages?

?

- The mean of a sample of data organized in a frequency distribution is computed by the following formula:

X values -

X-bar

Class midpoint

f - class

frequency

Sample

size

Sum of

frequencies

- A sample of twenty appliance stores in a large metropolitan area revealed the following number of VCR’s sold last week. Compute the mean number sold. The formula and computation is shown below.

= 325/20

= 16.25 VCR’s

- The table also gives the necessary computations.

USEFUL RELATIONSHIPS

If two averages of a moderately skewed

frequency distribution are known, the third

can be approximated. The formulas are:

Mode = Mean - 3(Mean - Median)

Mean = [3(Median) - Mode]/2

Median = [2(Mean) + Mode]/3

- TO COMPARE (COMPUTE) VARIOUS MEASURES OF DISPERSION FOR GROUPED AND UNGROUPED DATA.
- TO EXPLAIN THE CHARACTERISTICS, USES, ADVANTAGES, AND DISADVANTAGES OF EACH MEASURE OF DISPERSION.
- TO EXPLAIN CHEBYSHEV’S THEOREM AND THE EMPIRICAL (NORMAL) RULE.
- TO COMPUTE THE COEFFICIENTS OF VARIATION AND SKEWNESS.
- DEMONSTRATE COMPUTING STATISTICS WITH EXCEL.

MEASURES OF DISPERSION - UNGROUPED DATA

- Range:For ungrouped data, the rangeis the difference between the highest and lowest values in a set of data. To compute the range, use the following formula.
- EXAMPLE :A sample of five recent accounting graduates revealed the following starting salaries (in $1000): 17 26 18 20 19. The range is thus $26,000 - $17,000 = $9,000.

RANGE = HIGHEST VALUE - LOWEST VALUE

MEAN DEVIATION

- Mean Deviation:The arithmetic mean of the absolute values of the deviations from the arithmetic mean. It is computed by the formula below:

Individual

Value

Arithmetic

Mean

Sample

Size

The weights of a sample of crates ready for shipment to France are(in kg) 103, 97, 101, 106, and 103.

|103-102| +

|97-102| +

|101-102| +

|106 - 102| +

|103 - 102| = ?

1. X = 510/5 = 102 kg.

2. MD = 12/5 = 2.4 kg.

3. Typically, the

weights of the crates

are 2.4 kg from the

mean weight of 102 kg.

POPULATION VARIANCE

- Population Variance:The population variance for ungrouped data is the arithmetic mean of the squared deviations from the population mean. It is computed from the formula below:

Individual

value

Population mean

Sigma

square

Population

size

EXAMPLE

- The ages of all the patients in the isolation ward of Yellowstone Hospital are 38, 26, 13, 41, and 22 years. What is the population variance? The computations are given below.

m = S(X)/N

= 140/5 = 28.

s2 = S(X - m)2/N

= 534/5

= 106.8.

ALTERNATIVE FORMULA FOR THE POPULATION VARIANCE

Verify, using above formula, that the population

variance is 106.8 for the previous example.

Why would you use this formula?

THE POPULATION STANDARD DEVIATION

- Population Standard Deviation:The population standard deviation (s) is the square root of the population variance.
- For the previous example, the population standard deviation is s = 10.3344 (square root of 106.8).
- Note:If you are given the population standard deviation, just square that number to get the population variance.

- Sample Variance:The formula for the sample variance for ungrouped data is:

OR

Sample

variance

This sample variance is used to estimate the

population variance.

- Asampleof five hourly wages for blue-collar jobs is: 17 26 18 20 19. Find the variance.

= 100/5 = 20

X

s2 = 50/(5 - 1)

= 12.5.

SAMPLE STANDARD DEVIATION

- Sample Standard Deviation:The sample standard deviation (s) is the square root of the sample variance.
- For the previous example, the sample standard s = 3.5355 (square root of 12.5).
- Note:If you are given the sample standard deviation, just square that number to get the sample variance.

INTERPRETATION AND USES OF THE STANDARD DEVIATION

- Chebyshev’s theorem:For any set of observations (sample or population), the minimum proportion of the values that lie within k standard deviations of the mean is at least 1 - 1/k2, where k is any constant greater than 1.
- Empirical Rule:For any symmetrical, bell-shaped distribution, approximately 68% of the observations will lie within ± 1s of the mean (m); approximately 98% within ± 2s of the mean (m); and approximately 99.7% within ± 3s of the mean (m).

Bell-Shaped Curve showing the relationship between s and m.

Between:

1. 68.26%

2. 95.44%

3. 99.97%

m-3s

m-2s

m-1s

m

m+1s

m+2s

m+3s

INTERQUARTILE RANGE

- Interquartile range:Distance between the third quartile Q3 and the first quartile Q1.
- First Quartile:It is the value corresponding to the point below which 25% of the observations lie in an ordered data set.
- Third Quartile:It is the value corresponding to the point below which 75% of the observations lie in an ordered data set.

PERCENTILE RANGE

- Percentiles:Each data set has 99 percentiles, thus dividing the set into 100 equal parts. Note: in order to determine percentiles, you must first order the set.
- Percentile Range:The 10-to-90 percentile range is the distance between the 10th and 90th percentiles.

10-to-90 Percentile Range

10%

80%

10%

P10

P90

Max

Min

- Coefficient of Variation: The ratio of the standard deviation to the arithmetic mean, expressed as a percentage.
- For example, if the CV for the yield of two different stocks are 10 and 25. The stock with the larger CV has more variation relative to the mean yield. That is, the yield for this stock is not as stable as the other.

SKEWNESS

- Skewness:Measurement of the lack of symmetry of the distribution.
- The coefficient of skewness is computed from the following formula:
- Note: There are other coefficients of skewness.

Sk = 3(Mean - Median)/(Standard deviation)

RIGHT SKEWED DISTRIBUTION

MODE

MEDIAN

MEAN

Positively skewed

Mean and median

are to the RIGHT of

the mode.

=AVERAGE(A1:A10) Arithmetic Mean

=MEDIAN(A1:A10) Median Value

=MODE(A1:A10) Modal Value

=GEOMEAN(A1:A10) Geometric Mean

=QUARTILE(A1:A10,Q) Quartile Q Value

=MAX(A1:A10)-MIN(A1:A10) Range

=PERCENTILE(A1:A10,P) Percentile P Value

=AVEDEV(A1:A10) Mean Absolute Deviation (MAD)

=VAR(A1:A10) Sample Variance

=STDEV(A1:A10) Sample Standard Deviation

=VARP(A1:A10) Population Variance

=STDEVP(A1:A10) Population Standard Deviation

=STDEV(A1:A10)/AVERAGE(A1:A10) Coefficeint of

Variation

=SKEW(A1:A10) Coefficient of Skewness

(not same as book)

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