In this guide on descriptive statistics, we introduced you to the fundamental concepts of descriptive statistics. In this section, we’ll put those skills to the test with a few practice problems. Don’t worry if you’re having trouble remembering certain formulas or ideas, you can always compare your answers to the solutions posted below.

 

Practice Problems

 

Problem 1

A study has shown that males in the UK have steadily grown in height from the 1800s to 1980. Based on the data below, find the mean height for each sixty-year period. Then, choose an appropriate chart or plot to visualize this data.

YearHeight in Cm
1810169.7
1820169.1
1830166.7
1840166.5
1850165.6
1860166.6
1870167.2
1880168
1890167.4
1900169.4
1910170.9
1920171
1930173.9
1940174.9
1950176
1960176.9
1970177.1
1980176.8

 

Superprof

Problem 2

There are multiple data sources on the subject you are currently studying - weights of students in college. You’re interested in data that has low variability and a large sample size, the problem is that the data you found isn’t in kilograms but in pounds. Using variance and standard deviation in pounds (1 kg = 2.20462 lb), which data set out of the following would you choose?

MeasureData Set AData Set BData Set C
Sample Size15 6704 5009 334
Mean550464534
Standard Deviation432140210

 

Problem 3

You want to investigate the average amount of time people aged 20 to 40 spend on the phone each week. Interpret the chart below by finding the group mean of the hours spent on the phone.

AgeHours
20-2245
23-2536
26-2825
29-3116
32-3412
35-378.5
38-404

 

Histogram

 

Problem 4

You’re thinking about buying a new computer and are interested in looking at the price of computers on the market. Your budget is between 400 and 600 pounds. What percentage of the computers on the market are between your price range given the information below?

Mean Price of Computers on the Market540 pounds
SD of Computers on the Market120 pounds

 

Problem 5

You are studying the differences of the distributions of streams on a popular music streaming service called Dotify. You find the following chart in a report that studies the number of streams during the first quarter of the year. Interpret the chart using the table provided, measures of central tendency and variability. Keep in mind the data are in thousands.

JanuaryFebruaryMarch
Q0906585
Q112090115
Q2130100125
Q3140110135
Q4160125155

 

Multi boxplot 2

 

Problem 6

You are studying the frequency of the number of deaths each year for the top 10 causes of death, taken from the World Health Organization. Keep in mind that communicable diseases can pass from individual to individual. Given the following information, interpret the chart below.

Pie of pie

 

Cause of DeathTypeFrequency (in millions)
Ischaemic Heart DiseaseNon-communicable9433
StrokeNon-communicable5781
Chronic Obstructive Pulmonary DiseaseNon-communicable3041
Lower Respiratory InfectionsCommunicable2957
Alzheimer Disease and Other DementiasNon-communicable1992
Trachea, Bronchus, Lung CancersNon-communicable1708
Diabetes MellitusNon-communicable1599
Road InjuryInjury1402
Diarrhoeal DiseasesCommunicable1383
TuberculosisCommunicable1293

 

Solutions to Practice Problems

Solution Problem 1

Here, we need to:

  • Find the mean of each period
  • Plot this data

First, we find the mean by applying the formula for the mean,

    \[ \bar{x} = \frac{\Sigma x_{i}}{n} \]

YearHeight in Cm
1810 - 1860

    \[ \dfrac{169.7+169.1+166.7+166.5+165.6+166.6}{6} = \]

    \[ \dfrac{1004.2}{6} \]

    \[ = 167.4 \]

1870 - 1920

    \[ \dfrac{167.2+168+167.4+169.4+170.9+171}{6} = \]

    \[ \dfrac{1013.9}{6} \]

    \[ = 169 \]

1930 - 1980

    \[ \dfrac{173.9+174.9+176+176.9+177.1+176.8}{6} = \]

    \[ \dfrac{1055.6}{6} \]

    \[ = 175.9 \]

 

As we can see, the average height increases over time. This becomes even more apparent when we plot the data.

Bar chart 2

 

Solution Problem 2

Here, you were asked to:

  • Convert the variance and SD to pounds using the conversion 1 kg = 2.20462 lb
  • Choose a data set with low variability and a large sample size

To convert the variance and SD, we simply need to follow the rules for changing units, as seen in the table below.

MeasureData Set AData Set BData Set C
Sample Size15 6704 5009 334
Mean

    \[ \dfrac{550}{2.2} = 250 \]

    \[ \dfrac{464}{2.2} = 211 \]

    \[ \dfrac{534}{2.2} = 243 \]

Standard Deviation

    \[ \dfrac{432}{2.2} = 196 \]

    \[ \dfrac{140}{2.2} = 64 \]

    \[ \dfrac{210}{2.2} = 95 \]

CV79%30%39%

To find a preferred data set, you can use the coefficient of variation. Recall that the formula is,

[\

CV = \frac{s}{\bar{x}} *100%

\]

Which tells us the proportion of the standard deviation to the mean. This is what appears in the last row of the table. Because Data Set C has the second lowest variability but almost double the sample size of Data Set B, we’ll choose Data Set C.

 

Solution Problem 3

In this problem, in order to interpret the chart you where asked to,

  • Interpret the chart by finding the group mean of the hours spent on the phone

To find the group mean, you simply have to follow the formula,

    \[ x_{group} = \frac{\Sigma(f_{i}*x_{m})}{n} \]

AgeHoursx_{m}f_{i}*x_{m}
20-224521945
23-253624864
26-282527675
29-311630480
32-341233396
35-378.536306
38-40439156
Total146.53822

 

Plugging this into the formula, you get,

    \[ x_{group} = \frac{3822}{146.5} = 26.1 \]

Which means that the group average is ages 26-28. This can be seen in the chart below.

Histogram 2

 

Solution Problem 4

In this problem, you were asked to find:

  • The percentage of computers on the market in your price range

To do this, we first have to find the z-scores of the upper and lower limits of our budget. Then, we’ll look these z-scores up in the left-tail z-table to find the percentages these two points represent on a normal distribution.

You’re thinking about buying a new computer and are interested in looking at the price of computers on the market. Your budget is between 400 and 600 pounds. What percentage of the computers on the market are between your price range given the information below?

Z-ScoreValue
Lower Limit: 400 pounds

    \[ \dfrac{(400-540)}{120} = -1.17 \]

Upper Limit: 600 pounds

    \[ \dfrac{(600-540)}{120} = 0.5 \]

 

Recall that negative z-scores are found by simply taking 1 minus the absolute value of that z-score. Take a look at the image below for more clarification.

Right and left tail z-score

Because the distribution is symmetrical, we know that 1 - the right-tail probability is the same magnitude as the negative z-score. Find the z-score in the image below.

z-table

Finding the probability of 1.17 in the z-table, we get 0.87900, which gives us the negative probability of,

    \[ z_{-1.17} = 1 - 0.87900 = 0.12100 \]

While the z-score for 0.5, looking at the z-table, is 0.69146.

To find the interval of the two probabilities, we simply take the difference between the two. This can be clarified in the image below.

interval z-score distribution

The percentage, then, is,

    \[ 0.69146 - 0.12100 = 0.57046 \]

Which means about 57% of the computers on the market are within your budget.

 

Solution Problem 5

In this problem, you were asked to:

  • Interpret the chart

Find sample responses in the table below.

MeasureInterpretation
Q0January had the highest minimum streams, at 90,000
IQR = Q3 – Q1All months had the same IQR of 20,000, which is where 50% of the data lies
Q2February had the lowest median streams, with 110,000

 

Solution Problem 6

In this problem, you had to,

  • Interpret the chart given the data table

Find some sample responses in the table below

ResponseInterpretation
Ischaemic heart disease, stroke, chronic obstructive pulmonary disease, dementias, lung cancers and diabetes mellitus make up 77% of the top 10 causes of death77% of the top 10 causes of death are made up of non-communicable diseases
Road Injury makes up only 5% of the top 10 causes of deathInjury makes up only 5% of the top 10 causes of death in the world
Lower respiratory infections, diarrhoeal diseases and tuberculosis make up about 18% of the top 10 causes of deathCommunicable diseases make up 18% of the top 10 causes of death

 

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.

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