Question

For each of the following, state whether you expect the distribution to be symmetric, right skewed, or left skewed. Also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or IQR. Explain your reasoning.
Housing prices in a country where 25% of the houses cost below $350,000, 50% of the houses cost below $450,000, 75% of the houses cost below $1,000,000 and there are a meaningful number of houses that cost more than $6,000,000.
The distribution is expected to be:___________.
1. symmetric
2. left skewed
3. right skewed
A typical observation is best represented by the:_______.
1. median
2. mean
The variability in the observations is best measured by the:________.
1. IQR
2. standard deviation

Answer 1

Answer:

1. right skewed

2.median

3. IQR

Step-by-step explanation:

The distribution is expected to be:___________.3. right skewed

This is because most of the data lies on the right side of the observations . It has a left tail . Tail means the elongated side where the data is in lesser amount. If we analyze we see that 75% of the houses cost below $1,000,000 and  a few houses are in between the cost range of $1,000,000 -$6,000,000 .

So there is lesser data on the left side meaning the distribution is right skewed.

A typical observation is best represented by the:_______.1. median

Normally when the distribution is symmetric it is best represented by the mean but since this is a right skewed the median provides the best observation.

The variability in the observations is best measured by the:________1. IQR

The interquartile range is best for skewed distributions just like the median. The standard deviation tells the percentage of observations it falls from the mean but in skewed distributions it is the interquartile range that gives the  best measure of the variation from the median .The IQR  ranges from the low of Q2 to the high of Q3  which is mostly the middle part .

Answer 2

Answer:

Q₃ - Q₂ > Q₂ - Q₁

$550,000 > $100,000

Therefore, the distribution is expected to be right skewed.

Since the distribution is right-skewed, the median would best represent a typical observation in the data.

Since the distribution is right-skewed, The inter-quartile range (IQR) would be a better choice to represent the variability of data.

Step-by-step explanation:

Housing prices in a country where 25% of the houses cost below $350,000.

Which means that first quartile is Q₁ = $350,000

50% of the houses cost below $450,000.

Which means that second quartile is Q₂ = $450,000

75% of the houses cost below $1,000,000

Which means that third quartile is Q₃ = $1,000,000

Recall that the distribution will be symmetric if the following relation holds true

Q₃ - Q₂ = Q₂ - Q₁

Q₃ - Q₂ = $1,000,000 - $450,000 = $550,000

Q₂ - Q₁ = $450,000 - $350,000 = $100,000

Hence the distribution is not symmetric since

Q₃ - Q₂ ≠ Q₂ - Q₁

The distribution is right skewed since

Q₃ - Q₂ > Q₂ - Q₁

$550,000 > $100,000

Therefore, the distribution is expected to be right skewed.

Since the distribution is right-skewed, the median would best represent a typical observation in the data

Therefore, A typical observation is best represented by the median.

Since the distribution is right-skewed, The inter-quartile range (IQR) would be a better choice to represent the variability of data.

Therefore, the variability in the observations is best measured by the inter-quartile range (IQR).

Note:

For a symmetric distribution, the mean would best represent a typical observation in the data and the variability of observations would be best represented using the standard deviation.