central tendency

statistics central tendency indicates where the center of the distrubiton tends to be.

This is determiend by: median, mode, mean.

Which method you use to determine central tendency is based on the scale and shape of the data.

Scale:

• nominal
• ordinal
• interval
• ratio

e.g. mean is usually used with interval/ratio data.

If the data is skewed also plays a part.

Determining if something is approximately normal distribution

1. Construct a histogram of stem-and-leaf plot. If it’s normal, they’ll be a symmetrical mound.
2. Compute $\mu \pm \sigma ,\mu \pm 2\sigma ,\mu \pm 3\sigma$ . The 68, 95, and 100% of the data should be within each of those ranges respectively.
3. Determine the Inter Quartile Range (IQR) and standard deviation. If $\frac{IQR}{\sigma }\approx 1.3$, then it’s approximately normal.
4. Construct a normal probability plot for the data. If it’s normal, it’ll be an approximately straight line.

Normal probability plot

It’s a scatterplot wherein the ranked values are graphed on one axis and their expected z-score from the standard normal distribution on the other axis. This likely requires statistical software.

Definitions

[Kurtosis]

[“Kurtosis”] is a measure of the peakedness of a data distribution. (e.g. how steep is the center of a normal distribution).

Skew

If a normal distribution leans to the left, it’s [positively] skewed. If a normal distribution leans to the right, it’s [negatively] skewed.

[Remember: It ‘skews’ towards the direction of the longer tail]

[deviation]

[Deviation] is $X-\bar{X}$ and indicates the distance above or below the mean. The sum of these scores will cancel out to zero.