Statistics: Central Tendencies

Measures of Central Tendencies

• Statistics involves collecting, understanding, and making sense of data.
• Studying statistics helps to conduct research, read journals, develop analytical thinking, and stay informed.
• Measures of central tendency help find the middle or average of data.
• The "center" refers to numbers/figures landing in the middle of a dataset.

Common Measures

Mean

• The average is a useful feature.
• Calculated by adding all numbers in a dataset, then dividing by the count of elements.

Median

• The middle value in a list sorted from smallest to largest.
• The element in the middle after arranging the data set in order.

Mode

• Most frequently occurring value in the a data set.

Purpose

• Measures of central tendency give a general idea/outcome of a test or study.
• The average gauges the number of exam takers or study participants.

Population vs Sample Mean

• Sample mean is the arithmetic mean of random values from a population sample.
• Population mean is the true mean from an entire population.
• The symbol for sample mean is X (x bar).
• The symbol for population mean is μ (Greek term mu).
• Sample mean is easier to calculate, but less accurate than population mean
• Sample mean's standard deviation is denoted by s.
• Population mean's standard deviation is denoted by σ.
• A sample can measure the average weight of citizens in a city.
• In formulas: n is the number of individuals in the sample, N is the number in the population.
• The sample mean should approximate the population mean; accuracy increases with sample size.

Sigma Notation

• Σ means summation of numbers in a grouping.

Formulas

• The Sample Mean formula is given by:
  • x = (Σ xi) / n
  • n is the number of terms in the sample (sample size)
• Population Mean formula is given by:
  • μ = (Σ xi) / N
  • N is the number of terms in the population (population size)
• Sample mean is the sum of data values divided by the number of data items.

Weighted Mean

• Used to calculate the average value when data is presented differently than arithmetic/sample mean.

Weighted Mean Formula

• Weighted Mean = Σwx / Σw
  • Σ = summation
  • w = weights
  • x = values
  • or WX = (ΣW X) / (ΣW)
  • WX - the weighted mean
  • W - the weight
  • X - individual scores
• For example, to calculate a students final score use the formula =(25)(88)+(30)(71)+(10)(97)+(35)(90)/100 = 84.5

Geometric Mean

• Geometric Mean (GM) finds the central tendency of a number set by finding the product and the nth root.
• Geometric mean values describe the summary of whole data, tendencies are mean, median, mode, and range.

Geometric Mean Formulas

• GM= nth √(x1 × x2 × ... × xn) or
• GM = (x1 × x2 × ... × xn)^(1/n)
• The geometric mean of 2 and 8 is √(2x8) = √16 = 4

Median Calculation

• The median mark is in the middle, with an equal amount either side in the data set.
• If there are an even number of scores, the middle two scores must be averaged.
• Rearranging the data set (65,55,89,56,35,14,56,55,87,45) from smallest to largest becomes (14, 35, 45, 55, 55, 56, 56, 65, 87, 89)
• The 5th & 6th score from the set (14, 35, 45, 55, 55, 56, 56, 65, 87, 89) are then taken and averaged, giving a final median score of 55.5

Mode Calculation

• Mode is the the number in a list that occurs most.
• No repeating numbers, the data set has no mode.
• Find the mode through rearranging the list of numbers from least to greatest, and counting the total appearances of each one.
• The number that appears the most is determined to be listed as mode.
• In 3, 5, 7, 13, 3, 7, 9, 3 the ode is 3.

Changing Units Effects

• When values are multiplied by a constant, mean and median increase by the same factor.
• Adding 10 to each score with original mean of 5 and median of 6, results in a new mean of 15 and a median of 16.
• Multiplying each score by ten when the mean is 5 and the median is 6, the new mean is 50 and the median is 60.

Mean, Median, Mode of Ungrouped and Grouped Data.

Mean

• The average of observations is the sum of the values divided by the number of observations.
• The mean of the data is given by x = f1x1 + f2x2 + ... + fnxn/f1 + f2+........... + fn
• One example mean calculation of a formula gives the sum of (f₁.x₁)/Σ₁ = 1100/50 = 22

Median

• First arrange values of data sets in ascending order.
• When the set is on odd number, then the media is (n + 1)/2.
• When the set is an even number, then the median will be the average of the n/2th and the (n/2 + 1)th observation.

Calculating Median

• The formula to calculate the Median is is Me = 1 + {hx (N/2 - cf )/f}
  • 1 = lower limit of median class.
  • h =width of median class.
  • f = frequency of median class,
  • cf = cumulative frequency of the class preceding the median class.
  • N = ∑fi
• With a data set and formula (Median, Me = 1+ h{(N/2-cf)/f} = 24 + 8 {(40 - 34)/24} answer becomes Median Me = 26.

Mode

• Occurs most often and has the greatest data concentration.
• Modal Class, is a frequency distribution which has the greatest frequency.

Calculating Mode

• The formula for calculating the mode is:
  • Mo = xk +h{(fk - fk-1)/(2fk - fk-1 - fk+1)}
    • Xk = lower limit of the modal class interval.
    • fk = frequency of the modal class.
    • fk-1= frequency of the class preceding the modal class.
    • fk+1 = frequency of the class succeeding the modal class.
    • h=width of the class interval.
• Data and formula, Mode, Mo = 40 +10{(28-12)/(2*28-12-20)} = 46.67 Mode = 46.67
• The relationship between the mean, median and mode is: Mode = 3(Median) - 2(Mean)

Ungrouped data

• Mean x = Σx / n
• In ascending order M = ((n+1)/2) value of observation.
• Mode is that value of the observation which occurs maximum number of times.