Measures of central tendency

what is a measure of central tendency?

a stat that describes the centre or typical value of a distribution

mean, median, and mode

what is the mean?

the average and the formula is sum of all values/N

add the scores and divide by number of scores

what is the median and why is it useful?

the middle score are arranging values from smallest to largest

the median is useful when there are outliers

formula for odd numbers: (n)/2

formula for even numbers: (n+1)/2 → since it’s even there will be 2 people, look for their scores and average it to find the mean ONLY FOR A FREQUENCY DISTRIBUTION

what is the mode?

the most frequent number and the only measure that can be used for nominal data

when should you use each measure?

mean: numerical data when the average is useful and extreme values aren’t distorting

median: ordered data

mode: most common value

what is a weighted average?

an average that accounts for how much importance or weight each value has

1) multiply each value by its weight

2) sum the weighted values

3) divide by the sum of the weights

example: Imagine you run a survey asking 20 families how many pets they own. Instead of listing out all 20 individual answers, the data is grouped into a frequency table:

• 0 Pets (Value): 3 families (Weight/Frequency)

• 1 Pet (Value): 8 families (Weight/Frequency)

• 2 Pets (Value): 6 families (Weight/Frequency)

• 3 Pets (Value): 3 families (Weight/Frequency)

Step 1: Multiply each value (\(x\)) by its frequency (\(w\))

• 0 pets: \(0 \times 3 = 0\)

• 1 pet: \(1 \times 8 = 8\)

• 2 pets: \(2 \times 6 = 12\)

• 3 pets: \(3 \times 3 = 9\)

Step 2: Sum the weighted values

• (0 + 8 + 12 + 9 = 29 (This represents the total number of pets owned across all families)

Step 3: Sum the total weights

• (3 + 8 + 6 + 3 = 20 (The total sample size, \(n = 20\))

Step 4: Divide the sums

• 29/20 =1.45

The weighted statistical mean is 1.45 pets per family.

what is skewness and how does it affect the mean?

right-skewed: long tail towards larger values; the mean is often greater than the median

left-skewed: long tail towards smaller values; the mean is often less than the median

symmetrical: both sides are balanced; the mean and median are equal

means, medians, and skew

• median and mean aim to provide the typical value in that they describe the centrality

• when the distribution is fairly symmetrical it doesn’t include extreme cases (outliers), the mean and median will give similar (if not identical) values

• when the distribution is asymmetrical it includes extreme cases (outliers)

  • the mean could be misleading (artificially pulled towards the extreme)

  • the median is the preferred measure of central tendency