Measures of Central Tendency: Representative Value, Mean, Midian and Mode

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First, you can describe a group of scores in terms of a representative (or typical) value, such as an average. A representative value gives the central tendency of a group of scores

Representative Value

  • A representative value is a simple way, with a single number, to describe a group of scores (and there may be hundreds—or even thousands—of scores). The main representative value we consider is the mean.

  • describing how the numbers are spread out in a group of scores as amount of variation, or variability, among the scores. The two measures of variability you will learn about are called the variance and standard deviation

Measures of Central Tendency

The central tendency of a group of scores (a distribution) refers to the middle of the group of scores as mean, mode, and median

Mean

  • Usually the best measure of central tendency is the ordinary average, the sum of all the scores divided by the number of scores.

  • The mean of the 10 scores 17, 8, 8, 7, 3, 1, 6, 9, 3, 82 is 6 (the sum of 60 dreams divided by 10 students).

  • Balance distribution on log

  • Mathematically, you can think of the mean as the point at which the total distance to all the scores above that point equals the total distance to all the scores below that point

  • The mean is the average of the scores, the balance point. The mean can be a decimal number, even if all the scores in the distribution have to be whole numbers

  • Σ, the capital Greek letter sigma, is the symbol for “sum of.” It means “add up all the numbers for whatever follows.” Add up all the scores & Divide this sum by the number of scores

Weighted Mean

Weighted Mean is a statistical method which calculates the average by multiplying the weights with its respective mean and taking its sum.

Geometric Mean

It is defined as the arithmetic mean of the values taken on a log scale. It is also expressed as the nth root of the product of an observation.

Harmonic Mean

  • A.M × G.M =H.M2. Their relationship can also be illustrated using the inequality: A ≥ G ≥ H.

  • Harmonic mean - It is the reciprocal of the arithmetic mean of the observations.

Median

  • Median. If you line up all the scores from lowest to highest, the middle score is the median

  • The median is much less affected by the extreme score. The median of these five scores is .81—a value that is much more representative of most of the scores

  • Thus, using the median deemphasizes the one extreme time, which is probably appropriate. An extreme score like this is called an outlier. (reaction time)

  • Outlier was much higher than the other scores, but in other cases an outlier may be much lower than the other scores in the distribution

  • Even scores – mean of 2 different middle scores

  • Figure how many scores there are to the middle score by adding 1 to the number of scores and dividing by 2. For example, with 29 scores, adding 1 and dividing by 2 gives you 15. The 15th score is the middle score

  • If you have one middle score, this is the median. If you have two middle scores, the median is the average (the mean) of these two scores

Mode

  • The mode is another measure of central tendency. The mode is the most common single value in a distribution

  • In a perfectly symmetrical unimodal distribution, the mode is the same as the mean

  • mode can be a particularly poor representative value because it does not reflect many aspects of the distribution

  • You can change some values, but mode would be unaffected (but mean is affected)

Compare Mean, Mode and Median

Compare Mean, Mode and median

Compare Mean, Mode and Median

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Is left or negatively skewed

a normal curve

A Normal Curve

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  • For a normal curve, the highest point falls exactly at the midpoint of the distribution. This midpoint is the median value, since half of the scores in the distribution are below that point, and half are above it. The mean also falls at the same point because the normal curve is symmetrical about the midpoint, and every score in the left-hand side of the curve has a matching score on the right-hand side. So, for a perfect normal curve, the mean, mode, and median are always the same value.

  • mean is a fundamental building block for most other statistical techniques

Variability

variability of a distribution

Variability of a Distribution

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  • variability of a distribution as the amount of spread of the scores around the mean.

  • If the scores are mostly quite close to the mean, then the distribution has less variability than if the scores are further from the mean.

Variance

Standard Deviation

the scores across mean

The Scores Across Mean

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How Spread Out Are the Scores Across Mean

  • variance is the average of each score’s squared difference from the mean

  • Suppose one distribution is more spread out than another. The more spread out distribution has a larger variance because being spread out makes the deviation scores bigger.

  • variance is based on squared deviation scores, which do not give a very easy-to-understand sense of how spread out the actual, no squared scores are

  • The variance is the sum of squares divided by the number of scores.

  • Measures of variability, such as the variance and standard deviation, are heavily influenced by the presence of one or more outliers (extreme values) in a distribution.

  • The sum of the deviation scores is always 0 (or very close to 0, allowing for rounding error). Always sum the deviation scores. If they do not add up to 0, total again.

  • Note: A common mistake when figuring the standard deviation is to jump straight from the sum of squared deviations to the standard deviation(by taking the square root of the sum of squared deviations). Remember, before finding the standard deviation, first figure the variance (by dividing the sum of squared deviations by the number of scores, N). Then take the square root of the variance to find the standard deviation.

  • The scores in the number of dreams example were 7, 8, 8, 7, 3, 1, 6, 9, 3, 8, and we figured the standard deviation of the scores to be 2.57. Now imagine that one additional person is added to the study and that the person reports having 21 dreams in the past week. The standard deviation of the scores would now be 4.96; adding this single score almost doubled the size of the standard deviation.

  • figure an unbiased estimate of the population variance by slightly changing the ordinary variance formula. The ordinary variance formula is the sum of the squared deviation scores divided by the number of scores. The changed formula still starts with the sum of the squared deviation scores, but divides this by the number of scores minus 1. Dividing by a slightly smaller number makes the result slightly larger. Dividing by the number of scores minus 1 makes the variance you get just enough larger to make it an unbiased estimate of the population variance. (This unbiased estimate is our best estimate of the population variance. However, it is still an estimate, so it is unlikely to be exactly the same as the true population variance)

  • unbiased estimate of the population variance (S2) estimate of the population variance, based on sample scores, which has been corrected so that it is equally likely to overestimate or underestimate the true population variance; the correction used is dividing the sum of squared deviations by the sample size minus 1, instead of the usual procedure of dividing by the sample size directly

Standard Deviation

  • describe the spread of a group of scores is the standard deviation

  • The standard deviation is simply the square root of the variance

  • If the variance of a group of scores is 100, the standard deviation is 10. If the variance is 9, the standard deviation is 3.

  • The variance is about squared deviations from the mean

  • The standard deviation is the average amount that scores differ from the mean.