Understanding Statistics - Research Methods Part-4

Continuation of Part 3.

Central Tendency

• Given a Histogram, how would you choose a number/numbers that accurately represents the typical salary

  • The value at which the frequency is highest called the mode, and this certainly works in describing the distribution, the most common value is the mode.

  • Value in the middle is called the median and this will also work

  • Average is a statistic that rests at a specific spot in the middle of the distribution

• Mode

  • Mode occurs with the highest frequency
  • So what is the mode of [2, 5, 5, 9, 8, 3]
  • Answer = 5 because it occurs twice in the dataset
  • In case where there are thousands of data point in a histogram, then the mode is the range that occurred with the highest frequency, because we cannot see the individual values but we can see which bin has the highest frequency.
  • In case where the entire histogram is at same level then it is called Uniform Distribution, such distributions have no mode.
  • In case where there are two or more distinct clear trends will have more than one mode, making it a bimodal distribution.

• Mode Explained

  • Mode can be used to describe any type of data we have, whether it is numerical or categorical.
  • Not all scores in the dataset affect mode only the repeating ones.
  • If we take a lot of samples from the same population, the mode will be different for each sample.
  • Mode changes with change in bin size.
  • There is no equation for mode, there is a procedure to find the mode, but we cannot describe it with an equation, since it really depends on how we present the data.
• Mean
  • Sum of all the numbers divided by the total numbers
  • If data is [1, 2, 3, 4, 5] then mean = (1 + 2 + 3 + 4 + 5)/5
  • For sample we say x bar = Sigma of x divided by n (small n)
  • For sample we say mu = Sigma of x divided by N (capital N)
• Properties of Mean
  • All scores in the distribution affect the mean.
    • Think of mean as a pivot trying to keep the scale balanced, if we add/remove a score the scale will become off-balance and will have to be recalculated to re-balance it.
  • The mean can be described with a formula.
  • Many samples from the same population will have similar or roughly similar mean.
  • The mean of the sample can be used to make inferences about the population it came from.
  • The mean will change if we add an extreme value to the dataset.
    • This is known as outlier, these are the values that are unexpectedly different from the other observed values.
    • Outliers create skewed distributions by pulling the mean towards the outlier and this causes misleading average/mean.
• Median
  • Sort the data
  • Find the middle value of the data
  • Median of even numbers is calculated by
    • First sorting them
    • Then we select the two middle values
    • Take average of these two middle values
  • When data has outlier, the median does not get affected much by departures from the norm, this tendency of median is called robust

• Median Formula -For even values where X is the value and n is the position of the value - (X(n/2) + X(n/2+1))/2 find the two middle values and then find average of those two values -For odd values - X(n+1)/2

• Positively Skewed (High frequency towards Left)
  • Mode or highest frequency will be towards the left due to highest frequency being there
  • Mean will be pulled towards the right because of lot of smaller non repeating values are in right
  • Median will be in the middle of Mode and Mean
  • So Mode is less than Median which is less than Mean (Mode < Median < Mean)
• Normally Distributed (frequency in centre)
  • Mean will be equal to Median which will be equal to Mode (Mean = Median = Mode)
  • Mode will occur in the centre bin where the frequency is the highest.
  • But also since the distribution is symmetrical therefore the Mean and the Median will both occur pretty much right in the centre.
• Measure Of Centre
  • Mean:
    • Mean has simple equation
    • Mean will always change if any data value changes
    • Mean is not affected by change in bin size, it will always be the same, not matter how we visualize the data with the histogram
    • Mean is affected severely by outliers
    • Mean is not easy to find just by looking at the histogram
  • Median
    • Median does not has a simple equation
    • Median will not always change if any data value changes
    • Median is not affected by change in bin size
    • Median is not affected severely by outliers
    • Median is not easy to find just by looking at the histogram
  • Mode
    • Mode does not has a equation
    • Mode will not always change if any data value changes
    • Mode is affected by change in bin size
    • Mode is not affected severely by outliers
    • Mode is easy to find just by looking at the histogram, because it is the highest frequency
    • It can be used to describe categorical data, such as gender or country of origin
• In an introductory statistics course, the same number of students scored below 75% as above 75% on the final exam. What shape(s) could the distribution of final exam scores have? - This is another way of saying that 75% was the median score on the exam. All of these distributions can have a median of 75%. - Uniform - Normal - Bimodal - Positively Skewed - Negatively Skewed