Chapter 3: Numerical Data Description

• Chapter 3 focuses on describing data using numerical measures.
• Focus is on computing measures of middle, variability, and using numerical measures to describe data effectively.

Summary Measures

• Data can be described numerically through measures of center and location, other measures of location, and measures of variation.

Measures of Center and Location

• Measures of center and location include the mean, median, and mode, providing an overview of where the data is centered.

Mean (Arithmetic Average)

• The mean can be thought of as the balance point or center of mass of the data.
• To calculate the mean, sum the values and divide by the number of values.
• The population mean (μ) involves summing all values in the population (Σxᵢ) and dividing by the total number of data values (N).
• The sample mean (x̄) sums all values in the sample (Σxᵢ) and divides by the number of values in the sample (n).
• Extreme values (outliers) affect the mean.
• The mean is generally used for interval/ratio data.
• The mean is the most common measure of central tendency.
• The formula for the population mean is μ = (Σx) / N, where N is the number of data values.
• The average occupancy rate is found to be 15.125 rooms per week

Sample Mean Calculation Steps on a Sharp Calculator

• Clear the memory before each question by pressing [2nd F], [ALPHA], [0], then [0].
• Set the calculator to 4 decimal places by pressing [SET UP], [0], [0], then [4].
• Use [MODE] [1] [0] to display "STAT 0."
• Enter data using the [DATA] key {ENT}.
• To get the mean, press [ALPHA] [x-bar] {on the 4 key}, then the = key.
• For a set of professor salaries, the mean salary is $170,571.
• Solved problem 3.3, the mean is found to be 14.63

Median

• Used for ordinal and interval/ratio data
• The median is the center value that divides sorted data into two halves.
• If the number of data points is odd, the median is the middle number; if even, it is the average of the two middle numbers.
• The median is not affected by extreme values.
• A skewed distribution benefits from the median as a better measure of center.
• 50% of professors earn less than $155,000 (median salary).
• The ordered array for professor salaries, with an extreme value included, remains at $155,000

Alternative Calculation Methods

• Method 1: Calculate i = (n+1)/2 to find the median index point.
• If i is an integer, it corresponds to the median's position.
• If i is not an integer, the median is the average of the values in the integer positions below and above i.
• Method 2: Calculate i = (1/2)n as the Median Index Point.
• If i is an integer, the median is the average of the values in positions i and i+1.
• If i is not an integer, round up to the next integer to find the median position.
• A data array is given to be: 4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23 with n = 12, the median position is calculated.
• The median is the average of the 6th and 7th values: (11+12)/2 = 11.5
• By finding the median in problem 3.3, the value is equal to 14.35.

Mode

• The mode as a measure of central tendency is the value in a data set that occurs most frequently.
• The mode is used for either numerical or nominal (categorical) data, making it useful for nominal data.
• Datasets may lack a mode, or exhibit multiple modes if several values share the highest frequency.
• Extreme values do not affect the mode.
• In Example 3-6 a mode group found to be 2 groups of size 4.

Skewed Data

• In right-skewed data, the mean is larger than the median.
• In left-skewed data, the mean is smaller than the median.

Shape of a Distribution

• How data is distributed is described.
• Distributions can be symmetric or skewed.
• In a left-skewed distribution, the mean is less than the median
• In a symmetric distribution, the mean equals the median.
• In a right-skewed distribution, the median is less than the mean.

Descriptive Measures

• Descriptive measures have been summarized in figure 3-6
• The mean is used for ratio intervals and is sensitive to extremes.
• The median is used for ratio intervals and ordinal values and is not used on all data.
• The mode is used for ratio intervals, ordinal values, and nominal data, however may not reflect the center.

Other Measures of Location

• Percentiles and quartiles are the two other measures of data location.

Percentiles

• To calculate the ith position = p/100 * n
• If it is not an integer, round up to the next integer which corresponds to the pth percentile in the dataset.
• If it is an integer, the pth percentile is the means of the dataset.
• As an example, calculating the 60th percentile of 19 ordered values.

Quartiles

• Quartiles split the ranked data into 4 equal groups.
• The second quartile is typically known to be the 50th percentile.
• A first Quartile example needs to be shown.
• A tutorial question requires arranging values into an ordered array to find the mean (280.54), median (293), and mode (325).
• It requires the identification of a measure of central tendency and calculation of the 1st and 3rd quartiles (271, 317).
• Also, problem 3.7 needs to consider the following
• Determine the median scores
• Determine the 25th and 75th percentiles
• Determine the 60th percentile

Answers Given

• Median: 71.5. i = (20+1)/2 = 10.5 is not an integer. Median is the average of the 10th and 11th values; (70+73)/2=71.5.
• 25th percentile: i=25/100*20=5. i is 5, the 25th percentile average of the 5th and 6th values = (59+65)/2 = 62.
• 75th percentile: i =75/10020=15. As i is 15, the 75th percentile average of the 15th and 16th values = (81+82)/2 = 81.5 60th percentile: i = 60/10020=12. As i is 12, the 60th percentile average of the 12th and 13th values = (73+78)/2 = 75.5

Proportions

• π = the proportion of population having some characteristic
• Formula for population proportion π = (number of occurrences in the population) / (population size)
• Sample proportion (p) provides an estimate of N:
  • Formula for sample proportion p = (number of occurrences in the sample) / (sample size)
• A proportion is a special form of the arithmetic average when scoring occurrences with 1.
• Non occurrences use a 0, and proportion p is the arithmetic average of these scores.

Question

• In a telephone follow-up survey of a new advertising campaign, 45 out of 150 individuals contacted could recall the new advertising slogan associated with the product.
• Compute the proportion of people who could recall the new advertising slogan.
• Is this value the population parameter π or the sample statistic p?

Variation

• If all of the data are not the same value, a set of data exhibits variation.
• Measures of variation are measures of spread or variability of the data values.

Measures of Variation

• Measures of Variation include
  • Range
  • Variance
  • Standard Deviation
  • Coefficient of Variation

Range

• The simplest variation measure.
• A difference between the largest and smallest observations.
• The simple formula is: Range = Xmaximum - Xminimum. Example range calculation is 14 - 1 = 13
• The way in which data is distributed is simply ignored
• The range can be too sensitive to outcomes.
• The weak measure of variation uses too few values to indicate the variation.

Interquartile Range

• The Interquartile Range:
  • Some outlier problems are eliminated.
  • High and low-valued observations are eliminated.
  • A Interquartile range = 3rd quartile and 1st quartiles
  • Or the 75th and 25th percentiles. Example: interquartile range shown = 57 – 30 => 27 Problem 3-1 requests an IQR only

Variance

• The variance is the average of the squared distances of the data values from the mean.
• Sample variance: S2 = Σ(x₁ - x)² / n-1
• where: n = sample size s2 = sample variance Population variance: σ² = Σ(Χ₁ - μ) 2 / N N = population size population variance (sigma squared) For data (Fleetwood Mobile Home):X 15 15 - 25 = 25 25 -25 = 35 35 -25 = 20 20 - 25 = 30 30 - 25 =

Sharp calculator data to find the mean: 15 [Data] 25 [Data] 35 [Data] 20 [Data] 30 [Data]

Mode = 0, STAT Press [Alpha] [4] = Mean is 25 Press [Alpha] [6] = Standard deviation 7.07

Problems

• Data is given for 3-25 Part (c), 3-26 Parts (b and c), 3-29 number of typo errors from pages of a book Compute the standard deviation for these sample data.
• The number of times a population of business execs has been to the previous month; by: Computing variance standard deviation and assuming that data represent a sample instead of population By discussing the difference between values computed.

Answer

• 3-25c= 3.1251 3-26b= var= 3.1389, STD=1.7717 3-26c= var= 3.7667, STD=1.9408 - 3-29
• Range = 28 VAR = 92.75 STD = 9.6307 Q1=23.5, Q3=40.5. IQR = 17.

Comparing Standard Deviations

• Same mean, but different standard deviations.

Coefficient of Variation

• The coefficient of variation is the ratio of the standard deviation to the mean expressed as a percentage.
• The coefficient of variation is used to measure the relative variation in the data.
• The relative data set is that there is quite a big variability in the Lumber data

Formula

• S Population Population CV =