Find the mean, median and mode of the data set: 3, 5, 5, 7, 8, 9, 9, 9, 12.
Mean: \( \dfrac{67}{9}\approx7.44 \). Median: 8. Mode: 9.
Measures of Location & Spread
10 exam-style questions on mean, median, mode, variance, standard deviation and quartiles. Tap Show answer for full working.
3 Easy4 Medium3 Hard
The mean of 5 numbers is 12. Four of the numbers are 8, 10, 15 and 11. Find the fifth number.
Sum \( =5\times12=60 \). Fifth \( =60-(8+10+15+11)=60-44=16 \).
Calculate the range and interquartile range for: 2, 4, 5, 7, 9, 11, 12, 15, 18.
Range: \( 18-2=16 \). \( Q_1=5 \), \( Q_3=12 \), IQR \( =7 \).
The data 4, 6, 7, 9, 10, 12 have mean \( \bar{x} \). Find \( \sum(x-\bar{x})^2 \) and hence the variance and standard deviation (population formulae).
\( \bar{x}=8 \). Deviations: \( -4,-2,-1,1,2,4 \). Squared sum \( =16+4+1+1+4+16=42 \). Variance \( =\dfrac{42}{6}=7 \). SD \( =\sqrt{7} \).
A data set has \( n=20 \), \( \sum x=240 \), \( \sum x^2=3200 \). Find the mean and the standard deviation.
Mean \( =\dfrac{240}{20}=12 \). Variance \( =\dfrac{3200}{20}-12^2=160-144=16 \). SD \( =4 \).
The heights (cm) of 8 students are: 162, 158, 170, 165, 172, 160, 168, 175. Find the mean and sample standard deviation (divisor \( n-1 \)).
Mean \( =\dfrac{1330}{8}=166.25 \). \( \sum(x-\bar{x})^2\approx246.5 \). Sample variance \( \approx\dfrac{246.5}{7}\approx35.21 \). Sample SD \( \approx5.93 \) cm.
A frequency distribution for discrete data: \( x=1,2,3,4,5 \) with frequencies 4, 6, 8, 5, 2. Find the mean and the mode.
\( \sum f=25 \), \( \sum fx=1\cdot4+2\cdot6+3\cdot8+4\cdot5+5\cdot2=4+12+24+20+10=70 \). Mean \( =\dfrac{70}{25}=2.8 \). Mode \( =3 \).
For a set of data, \( \sum x=150 \), \( \sum x^2=2500 \), \( n=10 \). A new observation of 20 is added. Find the new mean and the new standard deviation (population).
Old mean \( =15 \). New mean \( =\dfrac{170}{11}\approx15.455 \). New \( \sum x^2=2500+400=2900 \). Variance \( =\dfrac{2900}{11}-\left(\dfrac{170}{11}\right)^2\approx263.636-238.843\approx24.793 \). SD \( \approx4.98 \).
The coded data \( y=\dfrac{x-50}{10} \) has mean \( \bar{y}=1.2 \) and SD \( s_y=0.8 \). Find the mean and standard deviation of \( x \).
\( \bar{x}=10\bar{y}+50=12+50=62 \). \( s_x=10s_y=8 \).
A data set of 12 observations has mean 25 and variance 16. Two additional observations, both equal to 25, are added. Show that the mean is unchanged and find the new variance.
New mean still 25. Old \( \sum(x-\bar{x})^2=12\times16=192 \). New sum of squares of deviations still 192 (extra terms 0). New variance \( =\dfrac{192}{14}\approx13.71 \).