Socy 10 Exam 1 Central Tendency

how to find the mode from a list of scores and a frequency distribution

-in a frequency distrib, the score with the largest frequency is the mode. -in a set of scores, count them all and the one that occurs the most is the mode

the mean is only for interval ratio variables

1) the mean is Appropriate only for interval-ratio variables. the variable must be able to assume consistent units/values for mathematical operations. (this is the theory. in reality, we often calculate the mean to tell us something about ordinal variables.)

median and levels of measurement

1) the median is Appropriate for interval-ratio or ordinal variables. must be able to order scores in order to find the median. for ordinal variables, it is almost always more appropriate than the mode. if the median falls between two cases for an ordinal variable with word labels rather than numerical values, express this in sentence form: "the median falls between ___ and ___" *the mean is almost always more desirable to use for interval variables, EXCEPT if there is a lot of skew (the mean and median are very different)

how to calculate the median

1)Order cases by score (low-to-high or low-to-high) 2)Find the Median Case (i.e., "median rank") MEDIAN CASE = (n+1) / 2 When you have an even number of cases, the above formula will gave a value ending in .5 Use the average of the two scores on either side (add the scores up and divide by two) to get the pint exactly in the middle of the distrib. the median is less affected by outliers than the mean, but sensitive to the number of scores (not their magnitudes).

mean= mathematical balancing point

2) the mean is the Mathematical balancing point of the distribution. in other words the magnitude of scores below the mean equal/balance the magnitude of scores above the mean. *NOT that there are the same number of cases above and below the mean; but rather that the total sum of all the score values is the same on both sides of the mean.

multiple modes

A distribution may have multiple modes. could be bimodal, trimodal, etc. the mode is less useful in these situations as a representation of central tendency. (it is no longer a single number to convey what is typical)

how to use measures of central tendency to find the shape of a distribution

Compare the mean and the median to each other. in a normal distribution, the mean and median are exactly identical. the more asymmetrical a distrib is, the farther apart eh mean and median scores will be. in a skewed distribution, the mean falls farther out along the tail, away from the bulk of the data. the median falls closer in to the bulk of the data.

positive vs negative skew

If the mean is larger than the mode, the distribution is positively skewed or skewed right (the tail goes to the right). If the mean is smaller than the median, the distrib is negatively skewed aka skewed left (the tail goes to the left).

mean

Mean = the arithmetic average of the scores in a distribution. the mean is equal to the sum of all scores divided by the number of scores in the distribution. the exact value of the mean does not necessarily have to appear as a score in the distribution

median

Median = The score of the case in the exact middle of the distribution. It is the Point in the distribution at which half of cases fall above that score and half of the cases fall below that score. Median = 50th percentile of the distribution

mode

Mode = The most frequently occurring score in the distribution. the mode is most in line with the "most typical" meaning of central tendency. the mode is the Single score most likely to occur = most likely to be selected at random out of the data set because it occurs the most frequently. (but does not take into account the distribution or magnitude of scores at all)

mean is sensitive to extremes

The mean is very sensitive to outliers (extremes in the data; data point that are way lower or higher than the bulk of the data). adding a score near the mean does not affect the mean very much, but adding one at the bottom or top of the distribution affects it a lot.

histogram

a histogram is a visualization of the shape of a distribution. they can be symmetrical (normal) or asymmetrical (skewed).

summary statistics/ summary measures

a summary measure summarizes a characteristic of a distribution (data set) using a SINGLE number/score (in contrast to a freq disturb or graphs, which show all the data). advantage: efficient disadvantage: may obscure info about the data two types of summary measures: 1)measures of central tendency (a single number used to convey what is typical, common, usual, or average in a distribution (mean median mode)) 2)measures of variability (single number used to convey the diversity or heterogeneity in a distribution (range, variance, SD))

mathematical implications of the balancing property:

a) The sum of the differences between the scores in a distribution and the mean of the distribution always equals zero. aka: Σ(Xi - x̄) = 0 b) The sum of the squared differences between the scores and the mean is smaller than the squared difference between the scores and any other point / value. Σ[(Xi - x̄)^2] = minimum c) scores in any distribution are more clustered around the mean than around any other point. d) in the absence of any other information, the mean is typically the best estimate of the score for any single case in the distribution.

how to calculate the mean from a frequency distribution

formula for calculating the mean fro m a frequency distribution: x̄ = (Σ f (Xi))/n f= the frequency of each specific score in the data set *multiply the scores with their frequencies, then sum those, then divide by n

how to calculate the mean

formula: x̄ = (Σ Xi)/n x̄ = symbol for the sample mean (the mean of variable x) Σ = summation sign, meaning add up the values of everything that follows it Xi= the individual values of the variable x (the scores of all the cases in the data set) n= total number of (valid) cases in sample

measures of central tendency

purpose: to convey what is typical, common, usual, or average in a distribution of scores using a single number. most common measures of central tendency: mean median mode

population mean vs. sample mean

sample mean is denoted by the symbol x with a bar over it ( x̄ ). population mean is denoted by the symbol mui ( μ ). sample mean is known and can be calculated from the data set. population mean is unknown and can be estimated from the data set using inferential statistics.

how to choose an appropriate measure of central tendency

three criteria for choosing which measure of central tendency to use: 1) level of measurement. nominal= only mode is valid ordinal= median and mode are both valid; almost always median is better interval= mean, median, and mode are all valid. almost always mean is better EXCEPT if data is skewed 2) shape of the distribution -the goal is to convey what is a common or typical score. since the mean is more sensitive to outliers, very skewed data with some extreme outliers can have a mean that is quite different from the most common/typical answer. if distrib is VERY skewed, use median instead. in all other cases use mean. 3) purpose -it is almost always the best to use the measure of central tendency at the highest level possible for your variable, unless you are specifically trying to express the most frequent score or the 50th percentile score.

how to calculate the median from a frequency distribution

two methods: Option 1: Find the median case in the Cumulative frequency column. median = (n+1/2). take this value and go down the cf column till you find the score that the median case falls within. Option 2: Find the 50th percentile in the C% column. the score of that column is the median.