Question

Question1: Data is collected for a continuous quantitative random variable. Summary statistics including measures of center and measures of spread/variation are calculated on the original raw data.
Assume that the original raw data is then ‘centered’ (i.e. the mean for the variable is subtracted from each raw score so that the new mean of the ‘centered’ data is at 0).
Show that the sample variance and, thus, the sample standard deviation is the same for the ‘centered’ data as it was for the original raw data.

Question2: Explain what information (if any) we can get from the following:
•! The mean of recoded nominal data with 2 outcomes e.g., Male = 0, Female = 1
•! The mean of recoded nominal data with 3 or more outcomes
e.g., White = 0, Black or African American = 1, Asian = 2, American Indian or Alaska Native = 3, Native Hawaiian or Other Pacific Islander = 4, Other = 5

Answer 1

1) Let,

X - be the continuous quantitative random variable.

And its mean is $\bar{X}$ and

Variance is $\sigma^2_{X} = \frac{\sum(x_{i} - \bar{X})^2}{N}$ ............................................ (1)

where x_i is the i^th observation of variable X and N is the number of observation.

If we center original raw data as the mean for the variable is subtracted from each raw score so that the new mean of the ‘centered’ data is at 0.

Let, we consider new variable for centered data is Y.

So y_i = x_i - $\bar{X}$ ....... is the i^th observation of new variable Y and N is the number of observation. Here N is same for raw and centered dataset.

Here we know that mean of variable Y is o. So variance is as

$\sigma^2_{Y} = \frac{\sum(y_{i} - \bar{Y})^2}{N} = \frac{\sum(y_{i})^2}{N} =\frac{\sum(x_{i} - \bar{X})^2}{N} = \sigma^2_{X}$................................. (2)

similarly for standard deviation . $\sigma_{Y} = \sigma_{X}$

Hence proved that the sample variance and, thus, the sample standard deviation is the same for the ‘centered’ data as it was for the original raw data.

2) Here given datasets are nominal values. We know that the nominal scale uses categories, so finding the mean or the median makes no sense. You could put the items in alphabetical order but even then, the middle item would have no meaning as a median. However, a mode (the most frequent item in the set) is possible.

For example i) In class room number of male or female is higher. and ii) In Pune University number of students of which country or color are most from White, Black or African American, Asian, American Indian or Alaska Native, Native Hawaiian or Other Pacific Islander, Other.