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4. a histogram shows that most students scored between 60–70 and 70–80,…

Question

  1. a histogram shows that most students scored between 60–70 and 70–80, with very few scoring below 40 or above 90.

a) describe the shape of the distribution.
b) predict whether the distribution has low variability or high variability, and justify your answer.

section c — box-and-whisker plots

  1. the five-number summary for class a is:

min = 20, q1 = 40, median = 55, q3 = 70, max = 95

a) draw the box-and-whisker plot.
b) calculate the iqr.
c) determine whether 95 is an outlier.
d) if the teacher says that class a’s scores have “moderate spread,” explain what part of the box plot supports this.

  1. comparing two box plots

class b: min = 55, q1 = 65, median = 72, q3 = 78, max = 82
class c: min = 40, q1 = 55, median = 63, q3 = 85, max = 100

a) which class has the greater iqr?
b) which class shows greater overall spread (using range)?
c) which class appears to have more consistent performance?
d) which class likely has the higher overall achievement? why?

Explanation:

Question 4a
Brief Explanations

A histogram with most data between 60 - 70 and 70 - 80, and few below 40 or above 90. The distribution is symmetric or approximately bell - shaped (normal - like) as the data is concentrated in the middle with tapering tails on both the lower (below 40) and upper (above 90) ends.

Brief Explanations

To determine variability, we look at the spread of data. The data has a range from below 40 to above 90, but most is in 60 - 80. However, since there are data points far from the main cluster (below 40 and above 90), the distribution has high variability. A distribution with low variability would have data concentrated very closely, but here we have a spread from below 40 to above 90 with a main cluster in the middle, so the overall spread is relatively large.

Brief Explanations

To draw a box - and - whisker plot:

  1. Draw a number line that covers the range from the minimum (20) to the maximum (95).
  2. Draw a box from Q1 (40) to Q3 (70). Inside the box, draw a line at the median (55).
  3. Draw whiskers from the box to the minimum (20) and maximum (95) values.

Answer:

The distribution is approximately symmetric (or bell - shaped/normal - like). The majority of data is in the middle (60 - 80 range) with relatively few data points in the lower (below 40) and upper (above 90) tails, indicating a symmetric spread around the center.

Question 4b