Univariate Data

Lesson

Now that we know how to calculate the mean, median, mode and range, we will take a look at how to find missing data values if we already know some of the statistics.

Let's launch right in with some examples.

The range for the following set of data, arranged in ascending order, is $10$10.

Calculate the missing value, $x$`x`.

$2$2 $4$4 $7$7 $10$10 $x$`x`

Think: Since we know the numbers are in ascending order and we know the range is $10$10, we can add the range onto the lowest value of $2$2 to get the highest value, $x$`x`.

Do:

$x$x |
$=$= | $2+10$2+10 |

$x$x |
$=$= | $12$12 |

The median for the following set of data, arranged in ascending order, is $5$5, and the range is $15$15.

Calculate the missing values of $x$`x`and $y$`y`.

$x$`x` $3$3 $y$`y` $7$7 $16$16

Think: First let's work with the range and find $x$`x`. This time $x$`x` is the smallest value, and the largest value is $16$16. We can subtract the range from the largest value to get the smallest value.

Do:

$x$x |
$=$= | $16-15$16−15 |

$x$x |
$=$= | $1$1 |

Think: Now let's use the median value to find $y$`y`. We know the median is the middle value, and in this set of $5$5 numbers the middle value is $y$`y`.

Do: $y=5$`y`=5

The following set of data, arranged in ascending order, has a mean of $4$4, a median of $3$3 and a mode of $1$1.

$1$1 $x$`x` $2$2 $y$`y` $z$`z` $10$10

Find the values of $x$`x`, $y$`y` and $z$`z`.

Think: We can either start by using the information about the mode or the median. Let's start with the mode.

Do: Since there needs to be a mode of $1$1, and the numbers are in ascending order, then $x=1$`x`=1

Think: Now we'll find the median. There are $6$6 data points, so the median will be the average of $2$2 and $y$`y`.

Do:

$3$3 | $=$= | $\frac{2+y}{2}$2+y2 |

$6$6 | $=$= | $2+y$2+y |

$6-2$6−2 | $=$= | $y$y |

$y$y |
$=$= | $4$4 |

Think: Lastly, we need to use the mean to find the value of $z$`z`.

Do:

$4$4 | $=$= | $\frac{1+1+2+4+z+10}{6}$1+1+2+4+z+106 |

$24$24 | $=$= | $18+z$18+z |

$24-18$24−18 | $=$= | $z$z |

$z$z |
$=$= | $6$6 |

Six numbers $6$6, $2$2, $7$7, $18$18, $17$17 and an unknown number $x$`x` have a median of $8.5$8.5.

Determine the value of $x$`x`

Write all the known numbers in increasing order.

(on one line with each number separated by a comma).

Determine the missing value $x$

`x`

Five numbers have a range of $16$16, a mode of $2$2, a median of $7$7 and a mean of $8$8. The minimum number in the set is $2$2.

From the information given, we can determine straight away the following 3 values

minimum median maximum $\editable{}$ ?? $\editable{}$ ?? $\editable{}$ We also know that the mode is $2$2 and the mean is $8$8. Using one of these pieces of information, fill in the 2nd value in the table.

minimum median maximum $2$2 $\editable{}$ $7$7 ?? $18$18 The final score will require a calculation. Find the final score $x$

`x`.Hence, complete the table.

minimum median maximum $2$2 $2$2 $7$7 $\editable{}$ $18$18

Four numbers have a range of $5$5, a median of $9$9 and a mode of $11$11. Write the four numbers of this data set.

(on one line separated by a comma)

Plan and conduct surveys and experiments using the statistical enquiry cycle:– determining appropriate variables and measures;– considering sources of variation;– gathering and cleaning data;– using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends in multivariate data sets;– comparing sample distributions visually, using measures of centre, spread, and proportion;– presenting a report of findings