Ch. 3 Numerical Descriptive Measures

Objectives

• Measures of central tendency
• Measures of variation and identifying outliers
• Measures of shape and relative location in numerical variables.
• Compute descriptive summary measures for a population.
• Interpret Empirical Rule
• Calculate the covariance and the coefficient of correlation.

Summary Definitions

• Central Tendency: The extent to which values of a numerical variable group around a typical or central value.
• Variation: The amount of dispersion or scattering away from a central value that the values of a numerical variable show.
• Shape: The pattern of the distribution of values from the lowest value to the highest value.

Measures of Central Tendency

• Arithmetic Mean
• Median
• Mode
• Geometric Mean (not exam material)

The Mean

• The arithmetic mean is the most common measure of central tendency.
• Each value plays an equal role, serving as a balance point in a data set. $\bar{X} = \frac{\sum<em>{i=1}^{n} X</em>i}{n}$
  • $\bar{X}$: Pronounced x-bar (mean of a sample)
  • $n$: Sample size
  • $X_i$: The ith value

Calculation and Impact

• Mean = sum of values divided by the number of values.
• Affected by extreme values (outliers).

Example
* Data Set 1: 11, 12, 13, 14, 15, 16, 17, 18, 19, 20; Mean = 15
* Data Set 2: 11, 12, 13, 14, 15, 16, 17, 18, 19, 100; Mean = 31.5

• The mean is a poor measure of central tendency in the presence of outliers.

The Median

• In an ordered array, the median is the “middle” number (50% above, 50% below).
• Less sensitive than the mean to extreme values. Example
  • Data Set 1: 11, 12, 13, 14, 15, 16, 17, 18, 19, 20; Median = 15.5
  • Data Set 2: 11, 12, 13, 14, 15, 16, 17, 18, 19, 100; Median = 15.5

Locating the Median

• Sort the values in numerical order (smallest to largest).
• Use the formula to find the position of the middle number: $\frac{n+1}{2}$
  • If $n$ is odd, the median is the middle number.
  • If $n$ is even, the median is the average of the two middle numbers.
    Example
  • Ranked values: 29, 31, 35, 39, 39, 40, 43, 44, 44, 52 (n = 10)
  • Median position: $\frac{10+1}{2} = 5.5$
  • Median = $\frac{39 + 40}{2} = 39.5$

The Mode

• Mode is the value that occurs most often.
• Not affected by extreme values.
• Used for either numerical or categorical data.
• There may be several modes in a data set or no mode at all. Example
  • Data Set 1: 29, 31, 35, 39, 39, 40, 43, 44, 44, 52; Modes = 39 and 44
    Data Set 2: No Mode

Review Example: House Prices

• House Prices: $2,000,000, $500,000, $300,000, $100,000, $100,000
• Mean: \frac{3,000,000}{5} = $600,000
• Median: $300,000
• Mode: $100,000

Which Measure to Choose?

• The mean is generally used unless extreme values (outliers) exist.
• The median is often used since it is not sensitive to extreme values.
• In many situations, it makes sense to report both the mean and the median.

Measures of Variation

• Measures of variation give information on the spread or variability or dispersion of the data values.
• Types:
  • Range
  • Variance
  • Standard Deviation
  • Coefficient of Variation

The Range

• Simplest measure of variation.
• Difference between the largest and the smallest values: $Range = X<em>{largest} – X</em>{smallest}$Example
  • Data: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
  • Range = 13 - 1 = 12

Why The Range Can Be Misleading

• Does not account for how the data are distributed.
• Sensitive to outliers. Examples
  • 7, 8, 9, 10, 11, 12; Range = 12 - 7 = 5
  • 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5; Range = 5 - 1 = 4
  • 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 120; Range = 120 - 1 = 119

The Sample Variance

• Measures the average scatter around the mean.
• Calculated as the average of squared deviations of values from the mean.
• Formula: $S^2 = \frac{\sum<em>{i=1}^{n} (X</em>i - \bar{X})^2}{n-1}$
  • $\bar{X}$ = arithmetic mean
  • $n$ = sample size
  • $X_i$ = ith value of the variable X

The Sample Standard Deviation

• The square root of the variance.
• Has the same units as that of the original sample data.
• Neither the variance nor the standard deviation can ever be negative.
• Sample standard deviation:
  $S = \sqrt{S^2}$

Steps for Computing Standard Deviation:

1. Compute the difference between each value and the mean.
2. Square each difference.
3. Add the squared differences.
4. Divide this total by n-1 to get the sample variance.
5. Take the square root of the sample variance to get the sample standard deviation.

Calculation Example

• Sample Data ($X_i$): 10, 12, 14, 15, 17, 18, 18, 24
• n = 8
• Mean = $\bar{X}$ = 16

Comparing Standard Deviations

• All data sets can have the same mean but different standard deviations.
• The more spread out the observations are, the larger the standard deviation.

Summary Characteristics

• The more the data are spread out, the greater the range, variance, and standard deviation.
• The more the data are concentrated, the smaller the range, variance, and standard deviation.
• If the values are all the same (no variation), all these measures will be zero.
• None of these measures are ever negative.

The Coefficient of Variation

• Measures relative variation.
• Always in percentage (%).
• Shows variation relative to mean.
• Can be used to compare the variability of two or more sets of data measured in different units.
• Formula:
  $CV = (\frac{S}{\bar{X}}) * 100$

Comparing Coefficients of Variation

Example
* Stock A: Mean price = $50, Standard deviation = $5, CV = 10%
* Stock B: Mean price = $100, Standard deviation = $5, CV = 5%

• Stock B is less variable relative to its mean price.

Example
* Stock A: Mean price = $50, Standard deviation = $5, CV = 10%
* Stock C: Mean price = $8, Standard deviation = $2, CV = 25%

• Stock C has a much smaller standard deviation but a much higher coefficient of variation.

Locating Extreme Outliers: Z-Score

• An extreme value or outlier is a value located far away from the mean.

• Z scores are useful in identifying outliers.

• The Z score of a value is the difference between that data value and the mean, divided by the standard deviation.

• Formula:
  $Z<em>i = \frac{X</em>i - \bar{X}}{S}$

  • $X_i$ represents the ith data value
  • $\bar{X}$ is the sample mean
  • $S$ is the sample standard deviation
    Example
  • Suppose a data value $X<em>i = 10$ from the data set with a sample mean $\bar{X}= 2$ and a sample standard deviation $S = 4$, then the Z score for $X</em>i$ is $Z = \frac{10 – 2}{4} = 2$

• A Z score of 0 indicates that the data value is the same as the mean.

  • If a Z score is a positive or negative number, it indicates whether the data value is above or below the mean and by how many standard deviations above or below the mean.

• The Z-score is the number of standard deviations a data value is away from the mean.

• A data value is considered an extreme outlier if its Z-score is less than -3.0 or greater than +3.0.

• The larger the absolute value of the Z-score, the farther the data value is from the mean.

Example
* Suppose the mean math SAT score is 490, with a standard deviation of 100.
* Compute the Z-score for a test score of 620 and determine if it is an outlier.
* $Z = \frac{620-490}{100} = 1.3$

• A score of 620 is 1.3 standard deviations above the mean and would not be considered an outlier.

Shape of a Distribution

• Describes how data are distributed.
• In a symmetrical distribution, the values below the mean are distributed exactly as the values above the mean.
• Two useful shape related statistics are:
  • Skewness: Measures the extent to which data values are not symmetrical.
  • Kurtosis (not test material): Measures the peakedness of the curve of the distribution.

Shape of a Distribution (Skewness)

• Measures the extent to which data is not symmetrical.
• Left-Skewed: Mean < Median, Skewness Statistic < 0
• Symmetric: Mean = Median, Skewness Statistic = 0
• Right-Skewed: Median < Mean, Skewness Statistic > 0

General Descriptive Stats Using Microsoft Excel Functions

• =AVERAGE(range)
• =MEDIAN(range)
• =MODE.SNGL(range)
• =STDEV.S(range)
• =VAR.S(range)
• =Z.SCORE(x, mean, stdev)

Numerical Descriptive Measures for a Population

• Descriptive statistics discussed previously described a sample, not the population.
• Summary measures describing a population, called parameters, are denoted with Greek letters.
• Important descriptive population parameters are the population mean, population variance, and population standard deviation.

Sample statistics versus population parameters

The mean µ

• The population mean is the sum of the values in the population divided by the population size, N. $μ = \frac{\sum<em>{i=1}^{N} X</em>i}{N}$
  • $μ$ = population mean
  • $N$ = population size
  • $X_i$ = ith value of the variable X

The Variance σ^2

• Average of squared deviations of values from the mean.
• Population variance: $σ^2 = \frac{\sum<em>{i=1}^{N} (X</em>i - μ)^2}{N}$
  • $μ$ = population mean
  • $N$ = population size
  • $X_i$ = ith value of the variable X

The Standard Deviation σ

• Most commonly used measure of variation.
• Shows variation about the mean.
• Is the square root of the population variance.
• Has the same units as the original data.
• Population standard deviation:
  $σ = \sqrt{σ^2}$

The Empirical Rule

• The empirical rule approximates the variation of data that are in a symmetric bell-shaped distribution.
• Approximately 68% of the data in a symmetric bell shaped distribution is within 1 standard deviation of the mean or $µ ± 1σ$.
• Approximately 95% of the data in a symmetric bell-shaped distribution lies within two standard deviations of the mean, or $µ ± 2σ$.
• Approximately 99.7% of the data in a symmetric bell-shaped distribution lies within three standard deviations of the mean, or $µ ± 3σ$.

Using the Empirical Rule

• Suppose that the variable Math SAT scores is bell-shaped with a mean of 500 and a standard deviation of 90. Then:
• Approximately 68% of all test takers scored between 410 and 590, ($500 ± 90$).
• Approximately 95% of all test takers scored between 320 and 680, ($500 ± 180$).
• Approximately 99.7% of all test takers scored between 230 and 770, ($500 ± 270$).
• The empirical rule helps measure how the values distribute above and below the mean and can help identify outliers.
• You can consider values not found in the interval $µ ± 3σ$ as outliers.
• Note: this rule also applies to the bell-shaped sample data sets (i.e., $±1s$ contains 68% of data, $2s$ for 95%, $3s$ for 99.7%)

Measures Of The Relationship Between Two Numerical Variables

• The Covariance
• The Coefficient of Correlation

The Covariance

• The covariance measures the direction of the linear relationship between two numerical variables (X & Y).
• The sample covariance: $cov(X,Y) = \frac{\sum<em>{i=1}^{n} (X</em>i - \bar{X})(Y_i - \bar{Y})}{n-1}$
  • $n$ = number of the pairs
• Only concerned with the directional relationship.
• No causal effect is implied.

Interpreting Covariance

• cov(X,Y) > 0: X and Y tend to move in the same direction.
• cov(X,Y) < 0: X and Y tend to move in opposite directions.
• $cov(X,Y) = 0$: X and Y are independent.
• The covariance has a major flaw: It is not possible to determine the relative strength of the relationship from the size of the covariance.

Coefficient of Correlation

• Measures the relative strength of the linear relationship between two numerical variables.
• Sample coefficient of correlation: $r = \frac{cov(X,Y)}{S<em>X S</em>Y}$
  • Where,
• No causal effect is implied.

Features of the Coefficient of Correlation

• The population coefficient of correlation is referred to as $ρ$.
• The sample coefficient of correlation is referred to as $r$.
• Either $ρ$ or $r$ have the following features:
  • Unit free
  • Range between –1 and 1
  • The closer to –1, the stronger the negative linear relationship.
  • The closer to 1, the stronger the positive linear relationship.
  • The closer to 0, the weaker the linear relationship.

Interpreting the Coefficient of Correlation

Example
* r = 0.733

• There is a relatively strong positive linear relationship between test score #1 and test score #2.
• Students who scored high on the first test tended to score high on second test.