Question

Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976–1977 through 2004–2005.

• μ = 1000 FTES

• median = 1,014 FTES

• σ = 474 FTES

• first quartile = 528.5 FTES

• third quartile = 1,447.5 FTES

• n = 29 years

How many standard deviations away from the mean is the median? Additional Information: The population FTES for 2005–2006 through 2010–2011 was given in an updated report. The data are reported here.

Short answer

The median differs from the mean by 0.03 standard deviations.

Step-by-step solution

To find

How many standard deviations away from the mean is the median.

Explanation

The mean and the median are central tendency measures in statistics. One of the measurements of dispersion is the standard deviation, which is used to show how much individuals in a group differ from the mean. The Z-score indicates how far a value is above or below the mean in terms of standard deviation. The z-score is calculated using the following formula:

$z=\frac{x-\mu }{\sigma }$

$$\begin{gathered} x=isanydatavalue \\ \mu =isthepopulationmean \\ \sigma =isthepopulations\tan darddeviation \end{gathered}$$

In the given example, we are given the below parameters about the population:

$$\begin{gathered} \mu =1000\mathrm{FTES} \\ median=1014FTES \\ \sigma =474FTES \\ firstquartile=528.5\mathrm{FTES} \\ thirdquartile=1447.5FTES \\ n=29years \end{gathered}$$

The z-score formula is now used to determine how many standard deviations distant from the mean the median is.

$$\begin{gathered} z=\frac{\text{Median}-\mu }{\sigma } \\ =\frac{1014-1000}{474} \\ z=0.03 \end{gathered}$$

Hence the median is 0.03 standard deviations above the mean.