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Chapter 3

# HSS 2381 Chapter Notes - Chapter 3: Central Tendency, Xm Satellite Radio, Descriptive Statistics

by OC253010

School

University of OttawaDepartment

Health SciencesCourse Code

HSS 2381Professor

Eleanor RiesenChapter

3This

**preview**shows pages 1-2. to view the full**8 pages of the document.**Chapter 3: Central Tendency Variability, and Relative Standing

- Both central tendency and variability can be expressed by indexes that are descriptive statistics

- Central Tendency: refers to location of typical data value which cluster around another variable

- Indexes of central tendency provide a single number to characterize a distribution

- Central tendency measures come from distribution center of data values = indicate "typical" where

data values cluster

- Popularly called an average

A) Central Tendency Indexes => 3 alternative indexes:

1. The mode

2. The median

3. The mean

1) The Mode: the soe alue that ous ost feue; the ost popula soe

- Ex: Age: 26 27 27 28 29 30 31 = Mode = 27

The Mode: Advantages

1. Can be used with data measured on any measurement level (including nominal level)

2. Eas to opute

3. Reflects an actual value in the distribution, = easy to understand

4. Useful he thee ae + popula soes i ultiodal distiutios

The Mode: Disadvantages

1. Ignores most information in the distribution

2. Tends to be unstable (value varies a lot from one sample to the next)

3. Some distributions may not have a mode (ex = 10, 10, 11, 11, 12, 12)

2) The Median: point that divides the distribution into equal half; 50% are below median & 50% above

Ex: Age: 26 27 27 28 29 30 31 = Median (Mdn) = 28

The Median: Advantages

1. Not influenced by outliers

2. Patiulal good ide of hat is tpial he distiutio is skeed

3. Eas to opute

4. Appropriate for ordinal level

The Median: Disadvantages

1. Doesn't take actual data values into account—only an index of position

2. Value of median not necessarily an actual data value = more difficult to understand vs mode

3) The Mean: the arithmetic average & most commonly used; Data values are summed & divided by N

Ex: Age: 26 27 27 28 29 30 31 => Mean = 28.3

- Preferred for interval- & ratio-level data

Equation:

M = ΣX ÷ N

Where:

M / X bar = sample mean

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Σ = the sum of

X = actual data values

N = number of people

μ = Population means:

The Mean: Advantages

1. The balance point in the distribution:

2. Sum of deviations above the mean always exactly balances those below it

3. Doesn't ignore any information

4. Most stable index of central tendency

5. Many inferential statistics are based on the mean

The Mean: Disadvantages

1. Sensitive to outliers

2. Gies a distoted ie of hat is tpial he data ae skeed

3. Value of mean is often not an actual data value

A) Central Tendency in Normal Distribution => all three indexes coincide

b) Central Tendency in Skewed Distributions => mean is pulled off ete i the dietio of the ske

• Positively skewed = higher mean vs mode/median; negatively skewed = lower mean

=

B) Variability:

- Variability: how spread out/dispersed data values are in distribution/how similar/diff ple r from each other on variable

- 2 distributions with the same mean & shape could have different dispersion/variability

1) High variability = A heterogeneous distribution (A)

2) Low variability = A homogeneous distribution (B)

Variability:

Indexes of Variability:

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1. Range

2. Interquartile range

3. Standard deviation

4. Variance

1) Range: difference between the highest & lowest value in the distribution

Ex: Weights (pounds): 110 120 130 140 150 150 160 170 180 190 =range is 80 (190 – 110)

Range: Advantages

1. Easy to compute

2. Readily understood = quick summary of distribution's variability

3. Provides useful information of distribution when extreme; of interest to readers of a report

Range: Disadvantages

1. Depends on only 2 scores & doesn't take all information into account => thus

2. Highly unstable = cuz based on only 2 values; fluctuates from sample to sample

3. Sensitive to outlier

4. Influenced by sample size

2) Interquartile Range (IQR): based on quartiles; range of scores within which middle 50% scores lie

- Lower quartile (Q1 ): Point below which 25% of scores lie

- Upper quartile (Q3 ): Point below which 75% of scores lie

- IQR = the distance between these 2 values Q1 & Q3 = formula = IQR = Q3 - Q1

- IQR: Weights (pounds): 110 120 130 140 150 150 160 170 180 190 = IQR = 45.0 (172.5 – 127.5)

Interquartile Range: Advantages

1. Reduces influenced by outliers & extreme scores regarding variablity

2. More stable vs range = based on middle-range cases instead of extreme ^ & uses more info

3. Important role to detect outliers

4. Appropriate as index of variability with ordinal measures

Interquartile Range: Disadvantages

1. Not particularly easy to compute

2. Not well understood

3. Doesn't take all values into account; rarely used

3) Standard Deviation:

- Standard Deviation (SD): index that conveys how much, on avg, scores in a distribution vary

- SDs based on difference btw every score & mean value = deviation scores (x)

Deviation scores (x) = subtract mean from each score = x = X - M

- Standard deviation ex: Weights (pounds): 110 120 130 140 150 150 160 170 180 190

- M = 150; 1st person: x = -40; last person: x = +40

- Deviation scores summed in a distribution always = 0 = Thus, to compute SD s, deviation scores must

be squared (x2 ) before being summed

SD Equation: SD = Square root of: Σx2 ÷ (N -1)

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