Question

Discuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center?

Short answer

$\text{The mean}>\text{median}>\text{mode indicates positive skewness.}$

Step-by-step solution

Formula used for Mean, Median and Mode

The formula for calculating mean in case of discrete series is:

Mean$=\frac{\sum fx}{N}$

Where:

$f=$is the frequencies

$x=$number of gold medals won

$N=$total of frequencies

The formula for calculating median in case of discrete series is:

Median $=$size of ${\left(\frac{N+1}{2}\right)}^{\text{th}}$term

The mode is one which has maximum frequency.

Explanation for a

The mean, median, and mode are central tendency measurements in statistics. The mean is calculated by dividing the total number of observations by the number of observations.

The median is the point at which the data is divided into two equal halves. In the data, the mode is the most commonly occurring observation.

Calculation for a

Explanation for a

Median $=$Size of ${\left(\frac{20+1}{2}\right)}^{\text{ith}}$item

$=$Size of $(10.5{)}^{\text{th}}$item

$$\begin{gathered} =3+0.5(4-3) \\ =3.5 \end{gathered}$$

The mode is given below:

Mode $=1.$. Because the maximum frequency is for the $x=1$.

We know that:

If Mean > Median > Mode, then the data is positively skewed.

If Mean < Median < Mode, then the data is negatively skewed.

If Mean = Median = Mode, then the data is symmetric.

Since Mean > Median > Mode, therefore, the given data is positively skewed.

Explanation for b

The mean, median, and mode are central tendency measurements in statistics. The mean is calculated by dividing the total number of observations by the number of observations.

Themedianisthepointatwhichthedataisdividedintotwoequalhalves.

In the data, the mode is the most commonly occurring observation.

Calculation for b

The data obtained from stem and leaf plot is:

$46,49,53,56,57,57,57,58,60,60,63,63,64,64,65,66,67,67,67,68,70,71,71,72,73,74,77,78,78,79,80,81,83,85,88,90,90,93,93$

The mean can be calculated as:

$\text{Mean}=\frac{46+49+53+56+57+57+\dots +93}{39}=\frac{2733}{39}=70.1$

The median is calculated as:

First we need to sort the data in ascending order as:

$$\begin{gathered} 46,49,53,56,57,57,57,58,60,60,63,63,64,64,65,66,67,67,67,68,70,71,71,72,73,74,77,78,78,79,80,81,83,85,88,90,90,93,93 \\ Median={\left(\frac{39+1}{2}\right)}^{th}item \\ ={\left(20\right)}^{th}item \\ =68 \end{gathered}$$

The mode is the most frequently occurring observation and in the given data, we clearly see $57$and $67$occur three times. Therefore, the given data has two modes and is called bimodal.

Mode$=57,67$

We know that:

If Mean > Median > Mode, then the data is positively skewed.

If Mean < Median < Mode, then the data is negatively skewed.

If Mean = Median = Mode, then the data is symmetric.

Since Mean > Median > Mode, therefore, the given data is positively skewed.

Explanation for c

The mean, median, and mode are central tendency measurements in statistics. The mean is calculated by dividing the total number of observations by the number of observations.

Themedianisthepointatwhichthedataisdividedintotwoequalhalves.

In the data, the mode is the most commonly occurring observation.

Calculation of c

Calculation for c

The modified are calculated to make the class interval exclusive and is done by subtracting the lower limit by $0.005$and adding the upper limit by $0.005$.

The median class is calculated as:

Median - class $=\frac{25}{2}=12.5$

Since, the cumulative frequency $12.5$lies between $14.995-19.995$. Therefore, the median class is $14.995-19.995$

Median $=14.995+\left(\frac{12.5-9}{7}\right)\times 5$

$$\begin{gathered} =14.995+2.5 \\ =17.495 \end{gathered}$$

The mode is calculated as:

Mode $$\begin{gathered} =3*17.495-2*16.095 \\ =20.295 \end{gathered}$$

We know that:

If Mean > Median > Mode, then the data is positively skewed.

If Mean < Median < Mode, then the data is negatively skewed.

If Mean = Median = Mode, then the data is symmetric.

Since Mean < Median < Mode, therefore, the given data is negatively skewed.